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$Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x over denominator square root of x plus 4 end root minus 2 end fraction equals$

4
$2 \sqrt{2}$
$\frac{1}{\sqrt{2}}$
$\sqrt{2}$

Hint:

We can apply L'Hopital's rule, also commonly spelled L'Hospital's rule, whenever direct substitution of a limit yields an indeterminate form. This means that the limit of a quotient of functions (i.e., an algebraic fraction) is equal to the limit of their derivatives.
In this question, we have to find value of $Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x over denominator square root of x plus 4 end root minus 2 end fraction$ .

The correct answer is: 4

$Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x over denominator square root of x plus 4 end root minus 2 end fraction$
We first try substitution:
$Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x over denominator square root of x plus 4 end root minus 2 end fraction equals$ $Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator 0 over denominator square root of 0 plus 4 end root minus 2 end fraction space equals space 0 over 0$
Since the limit is in the form 0 over 0, it is indeterminate—we don’t yet know what is it. We need to do some work to put it in a form where we can determine the limit.
$Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x over denominator square root of x plus 4 end root minus 2 end fraction$
$Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator begin display style fraction numerator d over denominator d x end fraction end style x over denominator fraction numerator d over denominator d x end fraction square root of x plus 4 end root minus 0 end fraction space space space space space space left parenthesis space A p p l y space L apostrophe H o p i t a l apostrophe s space r u l e right parenthesis Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator begin display style 1 end style over denominator begin display style fraction numerator 1 over denominator 2 square root of x plus 4 end root end fraction end style end fraction space equals Lt subscript x not stretchy rightwards arrow 0 end subscript space 2 square root of x plus 4 end root space p u t space x equals 0 space t h e n comma space 2 square root of x plus 4 end root space equals space 4$

We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means 0 over 0 or $fraction numerator plus-or-minus infinity over denominator plus-or-minus infinity end fraction.$

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4

The correct answer is: 4

We first try substitution:Since the limit is in the form 0 over 0, it is indeterminate—we don’t yet know what is it. We need to do some work to put it in a form where we can determine the limit.

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4
$2 \sqrt{2}$
$\frac{1}{\sqrt{2}}$
$\sqrt{2}$

Two wires of same diameter of the same material having the length $ell$ and 2 $ell$ If the force is applied on each, what will be the ratio of the work done in the two wires?

Two wires of same diameter of the same material having the length $ell$ and 2 $ell$ If the force is applied on each, what will be the ratio of the work done in the two wires?

What is the relationship between Young's modulus Y, Bulk modulus k and modulus of rigidity $eta$ ?

What is the relationship between Young's modulus Y, Bulk modulus k and modulus of rigidity $eta$ ?

A $2 m$ long rod of radius $1 c m$ which is fixed from one end is given a twist of $0.8$ radian. The shear strain developed will be

A $2 m$ long rod of radius $1 c m$ which is fixed from one end is given a twist of $0.8$ radian. The shear strain developed will be