Q105 / 160 · TS EAMCET 2022 (19 Jul Shift 2) · Mathematics

$\lim _{x \rightarrow 2}\left[\left(x^2-4 x+4\right) \cos \left(\frac{2}{x-2}\right)+\frac{x^2-4}{x^3-2 x-4}\right]=$

  • A
    0
  • B
    $\infty$
  • C
    1
  • Correct answer
    D
    $\frac{2}{5}$

Solution

$\begin{aligned} & \lim _{x \rightarrow 2}\left[\left(x^2-4 x+4\right) \cos \left(\frac{2}{x-2}\right)+\frac{x^2-4}{x^3-2 x-4}\right] \\ & \Rightarrow \lim _{x \rightarrow 2}\left(x^2-4 x+4\right) \cos \left(\frac{2}{x-2}\right)+\lim _{x \rightarrow 2} \frac{x^2-4}{x^3-2 x-4} \\ & \Rightarrow 0 \times[-1,1]+\lim _{x \rightarrow 2} \frac{(x-2)(x+2)}{(x-2)\left(x^2+2 x+2\right)} \Rightarrow 0+\frac{2}{5}=\frac{2}{5}\end{aligned}$

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

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