Maths-
General
Easy

Question

Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator left parenthesis 2 x minus 3 right parenthesis left parenthesis square root of x minus 1 right parenthesis over denominator 2 x squared plus x minus 3 end fraction equals

  1. 1 over 10
  2. fraction numerator negative 1 over denominator 10 end fraction
  3. 1 fifth
  4. fraction numerator negative 1 over denominator 5 end fraction

Hint:

In this question we will use L Hospital theorem open parentheses i. e. comma space fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction equals fraction numerator f apostrophe left parenthesis x right parenthesis over denominator g apostrophe left parenthesis x right parenthesis end fraction close parentheses and then put the value of x.

The correct answer is: fraction numerator negative 1 over denominator 10 end fraction


    Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator left parenthesis 2 x minus 3 right parenthesis left parenthesis square root of x minus 1 right parenthesis over denominator 2 x squared plus x minus 3 end fraction
equals Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator 2 x to the power of begin display style 3 over 2 end style end exponent minus 3 to the power of begin display style 1 half end style end exponent minus 2 x plus 3 over denominator 2 x squared plus x minus 3 end fraction
U sin g space L space H o s p i t a l space t h e o r e m comma space
equals Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 2 x to the power of begin display style 3 over 2 end style end exponent minus 3 to the power of begin display style 1 half end style end exponent minus 2 x plus 3 over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 2 x squared plus x minus 3 end fraction
equals Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator 3 square root of x minus begin display style fraction numerator 3 over denominator 2 square root of x end fraction end style minus 2 over denominator 4 x plus 1 end fraction
equals fraction numerator 3 cross times 1 space minus begin display style 3 over 2 end style cross times 1 minus 2 over denominator 4 cross times 1 space plus 1 end fraction
equals fraction numerator 3 minus begin display style 3 over 2 end style minus 2 over denominator 5 end fraction
equals fraction numerator begin display style fraction numerator 6 minus 3 minus 4 over denominator 2 end fraction end style over denominator 5 end fraction
equals fraction numerator negative 1 over denominator 10 end fraction

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    General
    Maths-

    Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 11 x cubed minus 3 x plus 4 over denominator 13 x cubed minus 5 x squared minus 7 end fraction

    Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 11 x cubed minus 3 x plus 4 over denominator 13 x cubed minus 5 x squared minus 7 end fraction
u sin g space L space H o s p i t a l space t h e o r e m comma space
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d over denominator d x end fraction end style 11 x cubed minus 3 x plus 4 over denominator begin display style fraction numerator d over denominator d x end fraction end style 13 x cubed minus 5 x squared minus 7 end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 33 x squared minus 3 over denominator 39 x squared minus 10 x end fraction
A g a i n comma space u sin g space L space H o s p i t a l space t h e o r e m comma space
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 33 x squared minus 3 over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 39 x squared minus 10 x end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 66 x over denominator 78 x minus 10 end fraction
A g a i n comma space u sin g space L space H o s p i t a l space t h e o r e m comma space
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 66 x over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 78 x minus 10 end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript 66 over 78
equals 66 over 78
equals 11 over 13

    Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 11 x cubed minus 3 x plus 4 over denominator 13 x cubed minus 5 x squared minus 7 end fraction

    Maths-General
    Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 11 x cubed minus 3 x plus 4 over denominator 13 x cubed minus 5 x squared minus 7 end fraction
u sin g space L space H o s p i t a l space t h e o r e m comma space
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d over denominator d x end fraction end style 11 x cubed minus 3 x plus 4 over denominator begin display style fraction numerator d over denominator d x end fraction end style 13 x cubed minus 5 x squared minus 7 end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 33 x squared minus 3 over denominator 39 x squared minus 10 x end fraction
A g a i n comma space u sin g space L space H o s p i t a l space t h e o r e m comma space
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 33 x squared minus 3 over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 39 x squared minus 10 x end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 66 x over denominator 78 x minus 10 end fraction
A g a i n comma space u sin g space L space H o s p i t a l space t h e o r e m comma space
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 66 x over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 78 x minus 10 end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript 66 over 78
equals 66 over 78
equals 11 over 13
    General
    Maths-

    In straight capital delta A B C, cot A plus cot invisible function application B plus cot invisible function application C equals

    space space space space c o t space A plus c o t space B plus space c o t space C space
equals fraction numerator cos space A over denominator sin A end fraction plus fraction numerator cos space B space over denominator sin space B space end fraction plus fraction numerator cos space C over denominator sin space C end fraction
equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction. fraction numerator b c over denominator 2 increment end fraction plus fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction. fraction numerator c a over denominator 2 increment end fraction plus fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction. fraction numerator a b over denominator 2 increment end fraction space space space space space open square brackets u sin g space cos i n e space f o r m u l a space a n d space a r e a space o f space t r i a n g l e space open parentheses increment equals fraction numerator a b space sin C over denominator 2 end fraction equals fraction numerator b c space sin space A over denominator 2 end fraction equals fraction numerator c a space sin space B over denominator 2 end fraction close parentheses close square brackets
equals fraction numerator b squared plus c squared minus a squared plus c squared plus a squared minus b squared plus a squared plus b squared minus c squared over denominator 4 increment end fraction
equals fraction numerator a squared plus b squared plus c squared over denominator 4 increment end fraction

    In straight capital delta A B C, cot A plus cot invisible function application B plus cot invisible function application C equals

    Maths-General
    space space space space c o t space A plus c o t space B plus space c o t space C space
equals fraction numerator cos space A over denominator sin A end fraction plus fraction numerator cos space B space over denominator sin space B space end fraction plus fraction numerator cos space C over denominator sin space C end fraction
equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction. fraction numerator b c over denominator 2 increment end fraction plus fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction. fraction numerator c a over denominator 2 increment end fraction plus fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction. fraction numerator a b over denominator 2 increment end fraction space space space space space open square brackets u sin g space cos i n e space f o r m u l a space a n d space a r e a space o f space t r i a n g l e space open parentheses increment equals fraction numerator a b space sin C over denominator 2 end fraction equals fraction numerator b c space sin space A over denominator 2 end fraction equals fraction numerator c a space sin space B over denominator 2 end fraction close parentheses close square brackets
equals fraction numerator b squared plus c squared minus a squared plus c squared plus a squared minus b squared plus a squared plus b squared minus c squared over denominator 4 increment end fraction
equals fraction numerator a squared plus b squared plus c squared over denominator 4 increment end fraction
    General
    Maths-

    Lt subscript x not stretchy rightwards arrow straight infinity end subscript space fraction numerator 8 vertical line x vertical line plus 3 x over denominator 3 vertical line x vertical line minus 2 x end fraction

    Lt subscript x not stretchy rightwards arrow straight infinity end subscript space fraction numerator 8 vertical line x vertical line plus 3 x over denominator 3 vertical line x vertical line minus 2 x end fraction
    open vertical bar x close vertical bar equals open curly brackets table attributes columnalign left end attributes row cell x space space space comma space x greater or equal than 0 end cell row cell negative x comma space x less than 0 end cell end table close
t a k i n g space x space p o s i t i v e comma
limit as x rightwards arrow infinity of fraction numerator 8 x plus 3 x over denominator 3 x minus 2 x end fraction equals limit as x rightwards arrow infinity of fraction numerator 11 x over denominator x end fraction equals 11

    Lt subscript x not stretchy rightwards arrow straight infinity end subscript space fraction numerator 8 vertical line x vertical line plus 3 x over denominator 3 vertical line x vertical line minus 2 x end fraction

    Maths-General
    Lt subscript x not stretchy rightwards arrow straight infinity end subscript space fraction numerator 8 vertical line x vertical line plus 3 x over denominator 3 vertical line x vertical line minus 2 x end fraction
    open vertical bar x close vertical bar equals open curly brackets table attributes columnalign left end attributes row cell x space space space comma space x greater or equal than 0 end cell row cell negative x comma space x less than 0 end cell end table close
t a k i n g space x space p o s i t i v e comma
limit as x rightwards arrow infinity of fraction numerator 8 x plus 3 x over denominator 3 x minus 2 x end fraction equals limit as x rightwards arrow infinity of fraction numerator 11 x over denominator x end fraction equals 11
    parallel
    General
    Maths-

    Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 plus cos squared space x over denominator x plus 2007 end fraction

    Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 plus cos squared space x over denominator x plus 2007 end fraction
U sin g space L space H o s p t a l space t h e o r e m comma space
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 2 plus cos squared space x over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x plus 2007 end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 space cos space x space sin space x over denominator 1 end fraction
equals 2

    Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 plus cos squared space x over denominator x plus 2007 end fraction

    Maths-General
    Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 plus cos squared space x over denominator x plus 2007 end fraction
U sin g space L space H o s p t a l space t h e o r e m comma space
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 2 plus cos squared space x over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x plus 2007 end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 space cos space x space sin space x over denominator 1 end fraction
equals 2
    General
    Maths-

    Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses

    space space space space Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses
equals space Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses cross times fraction numerator square root of x squared plus x end root plus x over denominator square root of x squared plus x end root plus x end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator x squared plus x minus x squared over denominator square root of x squared plus x end root plus x end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator x over denominator square root of x squared plus x end root plus x end fraction
D i v i d i n g space b y space x comma space
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 1 over denominator square root of 1 plus begin display style 1 over x end style end root plus 1 end fraction
equals fraction numerator 1 over denominator square root of 1 plus 1 end fraction
equals 1 half.

    Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses

    Maths-General
    space space space space Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses
equals space Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses cross times fraction numerator square root of x squared plus x end root plus x over denominator square root of x squared plus x end root plus x end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator x squared plus x minus x squared over denominator square root of x squared plus x end root plus x end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator x over denominator square root of x squared plus x end root plus x end fraction
D i v i d i n g space b y space x comma space
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 1 over denominator square root of 1 plus begin display style 1 over x end style end root plus 1 end fraction
equals fraction numerator 1 over denominator square root of 1 plus 1 end fraction
equals 1 half.
    General
    General

    Elastic potential and gravitational potential energies are equal all the time.

    • The main difference between gravitational potential energy and elastic potential energy is that the origin of gravitational potential energy is the gravitational forces acting between two massive bodies whereas the origin of elastic potential energy is the electrostatic forces between molecules that make up a material
    • An object possesses gravitational potential energy if it is positioned at a height above (or below) the zero height. An object possesses elastic potential energy if it is at a position on an elastic medium other than the equilibrium position

    Elastic potential and gravitational potential energies are equal all the time.

    GeneralGeneral
    • The main difference between gravitational potential energy and elastic potential energy is that the origin of gravitational potential energy is the gravitational forces acting between two massive bodies whereas the origin of elastic potential energy is the electrostatic forces between molecules that make up a material
    • An object possesses gravitational potential energy if it is positioned at a height above (or below) the zero height. An object possesses elastic potential energy if it is at a position on an elastic medium other than the equilibrium position
    parallel
    General
    Maths-

    In straight triangle A B C comma cot A plus cot invisible function application B plus cot invisible function application C equals

    space space space space c o t space A plus c o t space B plus space c o t space C space
equals fraction numerator cos space A over denominator sin A end fraction plus fraction numerator cos space B space over denominator sin space B space end fraction plus fraction numerator cos space C over denominator sin space C end fraction
equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction. fraction numerator b c over denominator 2 increment end fraction plus fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction. fraction numerator c a over denominator 2 increment end fraction plus fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction. fraction numerator a b over denominator 2 increment end fraction space space space space space open square brackets u sin g space cos i n e space f o r m u l a space a n d space a r e a space o f space t r i a n g l e space open parentheses increment equals fraction numerator a b space sin C over denominator 2 end fraction equals fraction numerator b c space sin space A over denominator 2 end fraction equals fraction numerator c a space sin space B over denominator 2 end fraction close parentheses close square brackets
equals fraction numerator b squared plus c squared minus a squared plus c squared plus a squared minus b squared plus a squared plus b squared minus c squared over denominator 4 increment end fraction
equals fraction numerator a squared plus b squared plus c squared over denominator 4 increment end fraction

    In straight triangle A B C comma cot A plus cot invisible function application B plus cot invisible function application C equals

    Maths-General
    space space space space c o t space A plus c o t space B plus space c o t space C space
equals fraction numerator cos space A over denominator sin A end fraction plus fraction numerator cos space B space over denominator sin space B space end fraction plus fraction numerator cos space C over denominator sin space C end fraction
equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction. fraction numerator b c over denominator 2 increment end fraction plus fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction. fraction numerator c a over denominator 2 increment end fraction plus fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction. fraction numerator a b over denominator 2 increment end fraction space space space space space open square brackets u sin g space cos i n e space f o r m u l a space a n d space a r e a space o f space t r i a n g l e space open parentheses increment equals fraction numerator a b space sin C over denominator 2 end fraction equals fraction numerator b c space sin space A over denominator 2 end fraction equals fraction numerator c a space sin space B over denominator 2 end fraction close parentheses close square brackets
equals fraction numerator b squared plus c squared minus a squared plus c squared plus a squared minus b squared plus a squared plus b squared minus c squared over denominator 4 increment end fraction
equals fraction numerator a squared plus b squared plus c squared over denominator 4 increment end fraction
    General
    Maths-

    text  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript left parenthesis square root of x plus 1 end root minus square root of x right parenthesis

    text  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript left parenthesis square root of x plus 1 end root minus square root of x right parenthesis
equals text  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript left parenthesis square root of x plus 1 end root minus square root of x right parenthesis cross times fraction numerator left parenthesis square root of x plus 1 end root plus square root of x right parenthesis over denominator left parenthesis square root of x plus 1 end root plus square root of x right parenthesis end fraction
equals straight c fraction numerator x plus 1 minus x over denominator left parenthesis square root of x plus 1 end root plus square root of x right parenthesis end fraction
text = Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 1 over denominator square root of x open parentheses square root of 1 plus begin display style 1 over x end style end root text +1 end text close parentheses end fraction
equals space 1 over infinity cross times fraction numerator 1 over denominator square root of 1 plus begin display style 1 over infinity end style end root plus 1 end fraction
equals 0 cross times 1 over 1
equals 0.

    text  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript left parenthesis square root of x plus 1 end root minus square root of x right parenthesis

    Maths-General
    text  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript left parenthesis square root of x plus 1 end root minus square root of x right parenthesis
equals text  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript left parenthesis square root of x plus 1 end root minus square root of x right parenthesis cross times fraction numerator left parenthesis square root of x plus 1 end root plus square root of x right parenthesis over denominator left parenthesis square root of x plus 1 end root plus square root of x right parenthesis end fraction
equals straight c fraction numerator x plus 1 minus x over denominator left parenthesis square root of x plus 1 end root plus square root of x right parenthesis end fraction
text = Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 1 over denominator square root of x open parentheses square root of 1 plus begin display style 1 over x end style end root text +1 end text close parentheses end fraction
equals space 1 over infinity cross times fraction numerator 1 over denominator square root of 1 plus begin display style 1 over infinity end style end root plus 1 end fraction
equals 0 cross times 1 over 1
equals 0.
    General
    maths-

    Lt subscript x not stretchy rightwards arrow straight infinity end subscript space fraction numerator a subscript 0 plus a subscript 1 x to the power of 1 plus a subscript 2 x squared plus horizontal ellipsis plus a subscript n x to the power of n over denominator b subscript 0 plus b subscript 1 x to the power of 1 plus b subscript 2 x squared plus midline horizontal ellipsis plus b subscript m x to the power of w end fraction Where a subscript n greater than 0 comma b subscript m greater than 0 and n greater than m

    Lt subscript x not stretchy rightwards arrow straight infinity end subscript space fraction numerator a subscript 0 plus a subscript 1 x to the power of 1 plus a subscript 2 x squared plus horizontal ellipsis plus a subscript n x to the power of n over denominator b subscript 0 plus b subscript 1 x to the power of 1 plus b subscript 2 x squared plus midline horizontal ellipsis plus b subscript m x to the power of w end fraction Where a subscript n greater than 0 comma b subscript m greater than 0 and n greater than m

    maths-General
    parallel
    General
    physics-

    A ray of light is incident on an equilateral glass prism placed on a horizontal table For minimum deviation which of the following is true?

    A ray of light is incident on an equilateral glass prism placed on a horizontal table For minimum deviation which of the following is true?

    physics-General
    General
    physics-

    Two identical thin isosceles prisms of refracting angle ‘A’ and refractive index m are placed with their bases touching each other Two parallel rays of light are incident on this system as shown The distance of the point where the rays converge from the prism is

    Two identical thin isosceles prisms of refracting angle ‘A’ and refractive index m are placed with their bases touching each other Two parallel rays of light are incident on this system as shown The distance of the point where the rays converge from the prism is

    physics-General
    General
    physics-

    A ray of light strikes a plane mirror at an angle of incidence 450 as shown in the figure After reflection, the ray passes through a prism of refractive index 1.50, whose apex angle is 40 The angle through which the mirror should be rotated if the total deviation of the ray is to be 900 is

    A ray of light strikes a plane mirror at an angle of incidence 450 as shown in the figure After reflection, the ray passes through a prism of refractive index 1.50, whose apex angle is 40 The angle through which the mirror should be rotated if the total deviation of the ray is to be 900 is

    physics-General
    parallel
    General
    physics-

    A given ray of light suffers minimum deviation in an equilateral prism P Additional prisms Q and R of identical shape and of the same material as P are now added as shown in the figure The ray will now suffer

    A given ray of light suffers minimum deviation in an equilateral prism P Additional prisms Q and R of identical shape and of the same material as P are now added as shown in the figure The ray will now suffer

    physics-General
    General
    maths-

    In straight triangle A B C comma 4 R straight capital delta sin A over 2 sin invisible function application B over 2 sin invisible function application C over 2 equals

    In straight triangle A B C comma 4 R straight capital delta sin A over 2 sin invisible function application B over 2 sin invisible function application C over 2 equals

    maths-General
    General
    Maths-

    Lt subscript x not stretchy rightwards arrow straight infinity end subscript open square brackets square root of x squared plus a x plus b end root minus x close square brackets

    Lt subscript x not stretchy rightwards arrow straight infinity end subscript open square brackets square root of x squared plus a x plus b end root minus x close square brackets
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator open square brackets square root of x squared plus a x plus b end root minus x close square brackets open square brackets square root of x squared plus a x plus b end root plus x close square brackets over denominator open square brackets square root of x squared plus a x plus b end root plus x close square brackets end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator x squared plus a x plus b minus x squared over denominator open square brackets square root of x squared plus a x plus b end root plus x close square brackets end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator a x plus b over denominator open square brackets square root of x squared plus a x plus b end root plus x close square brackets end fraction
d i v i d i n g space t h e space n u m e r a t o r space a n d space d e n o m i n a t o r space b y space x comma
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator a plus begin display style b over x end style over denominator open square brackets square root of 1 plus begin display style a over x end style plus begin display style b over x squared end style end root plus 1 close square brackets end fraction
equals fraction numerator a over denominator square root of 1 plus 1 end fraction
equals a over 2

    Lt subscript x not stretchy rightwards arrow straight infinity end subscript open square brackets square root of x squared plus a x plus b end root minus x close square brackets

    Maths-General
    Lt subscript x not stretchy rightwards arrow straight infinity end subscript open square brackets square root of x squared plus a x plus b end root minus x close square brackets
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator open square brackets square root of x squared plus a x plus b end root minus x close square brackets open square brackets square root of x squared plus a x plus b end root plus x close square brackets over denominator open square brackets square root of x squared plus a x plus b end root plus x close square brackets end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator x squared plus a x plus b minus x squared over denominator open square brackets square root of x squared plus a x plus b end root plus x close square brackets end fraction
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator a x plus b over denominator open square brackets square root of x squared plus a x plus b end root plus x close square brackets end fraction
d i v i d i n g space t h e space n u m e r a t o r space a n d space d e n o m i n a t o r space b y space x comma
equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator a plus begin display style b over x end style over denominator open square brackets square root of 1 plus begin display style a over x end style plus begin display style b over x squared end style end root plus 1 close square brackets end fraction
equals fraction numerator a over denominator square root of 1 plus 1 end fraction
equals a over 2
    parallel

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