Maths-
General
Easy

Question

Lim subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x tan space 2 x minus 2 x tan space x over denominator left parenthesis 1 minus cos space 2 x right parenthesis squared end fraction equals

  1. 2
  2. -2
  3. 1 half
  4. fraction numerator negative 1 over denominator 2 end fraction

hintHint:

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 Lim subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x tan space 2 x minus 2 x tan space x over denominator left parenthesis 1 minus cos space 2 x right parenthesis squared end fraction.

The correct answer is: 1 half


    Lim subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x tan space 2 x minus 2 x tan space x over denominator left parenthesis 1 minus cos space 2 x right parenthesis squared end fraction
    Lim subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x tan space 2 x minus 2 x tan space x over denominator left parenthesis 1 minus cos space 2 x right parenthesis squared end fraction equals space Lim subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x tan space 2 x over denominator left parenthesis 1 minus cos space 2 x right parenthesis squared end fraction space minus Lim subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator 2 x tan space x over denominator left parenthesis 1 minus cos space 2 x right parenthesis squared end fraction
left curly bracket space W e space k n o w space t h a t space space space 1 minus cos 2 x space equals space 2 sin squared x space space right curly bracket
space Lim subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x squared tan space 2 x over denominator left parenthesis space 2 sin to the power of 4 x right parenthesis 2 x space end fraction space minus Lim subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x squared tan space x over denominator space 2 x sin to the power of 4 x space end fraction
left curly bracket space space A l s o space W e space k n o w space t h a t space space Lim subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator tan space x over denominator space x space end fraction space equals space 1 comma space Lim subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator sin space x over denominator space x space end fraction space equals space 1 space right curly bracket
space Lim subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x squared over denominator space 2 sin to the power of 4 x space end fraction space minus Lim subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x squared over denominator space 2 sin to the power of 4 x space end fraction space equals 0

    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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