Q105 / 200 · WBJEE 2014 · Mathematics

Let $f(x)$ be a differentiable function and $f^{\prime}(4)=5 .$ Then, $\lim _{x \rightarrow 2} \frac{f(4) - f\left(x^{2}\right)}{x-2}$ equals

  • A
    0
  • B
    5
  • C
    20
  • Correct answer
    D
    -20

Solution

Given, $f^{\prime}(4)=5$ $$ \begin{array}{ll} \text { Now, } \lim _{x \rightarrow 2} \frac{f(4)-f\left(x^{2}\right)}{x-2} & {\left[\frac{0}{0} \text { form }\right]} \\ =\lim _{x \rightarrow 2} \frac{0-f^{\prime}\left(x^{2}\right) \cdot 2 x}{1} \\ =\frac{-f^{\prime}(4) \cdot 2 \times 2}{1} \\ =-(5) \times 4=-20 \end{array} $$

Hence, the correct answer is option: 'D'.

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