Q72 / 150 · MHT CET 2007 · Mathematics

The value of $\lim _{x \rightarrow \infty}\left(\frac{x^{2}-2 x+1}{x^{2}-4 x+2}\right)^{x}$ is

  • Correct answer
    A
    $e^{2}$
  • B
    $e^{-2}$
  • C
    $e^{6}$
  • D
    None of these

Solution

$\begin{aligned} & \lim _{x \rightarrow \infty}\left(\frac{x^{2}-2 x+1}{x^{2}-4 x+2}\right)^{x} \\ &=\lim _{x \rightarrow \infty}\left(1+\frac{2 x-1}{x^{2}-4 x+2}\right)^{x} \\ &=e^{\lim _{x \rightarrow-\infty}\left(\frac{x(2 x-1)}{x^{2}-4 x+2}\right)}=e^{2} \end{aligned}$

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

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