Q99 / 160 · AP EAMCET 2024 (21 May Shift 1) · Mathematics

If $f(x)=\frac{5 x \cdot \operatorname{cosec}(\sqrt{x})-1}{(x-2) \operatorname{cosec}(\sqrt{x})}$, then $\lim _{x \rightarrow \infty} f\left(x^2\right)=$

  • A
    1
  • B
    -1
  • Correct answer
    C
    5
  • D
    -5

Solution

$\begin{aligned} & \text { } f(x)=\frac{5 x \operatorname{cosec}(\sqrt{x})-1}{(x-2) \operatorname{cosec}(\sqrt{x})} \\ & f\left(x^2\right)=\frac{5 x^2 \operatorname{cosec} x-1}{\left(x^2-2\right) \operatorname{cosec} x}=\frac{5 x^2}{x^2-2}-\frac{\sin x}{x^2-2} \\ & \lim _{x \rightarrow \infty} f\left(x^2\right)=\lim _{x \rightarrow \infty}\left[\frac{5 x^2}{x^2-2}-\frac{\sin x}{x^2-2}\right] \\ & =\lim _{x \rightarrow \infty}\left[\frac{10 x}{2 x}-0\right]=5\end{aligned}$

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

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