Q81 / 150 · MHT CET 2021 (22 Sep Shift 2) · Mathematics

$\lim _{x \rightarrow 2}(x-1)^{\frac{1}{3 x-6}}=$

  • A
    $\mathrm{e}^2$
  • B
    $\mathrm{e}^3$
  • Correct answer
    C
    $\mathrm{e}^{\frac{1}{3}}$
  • D
    $\mathrm{e}^{\frac{1}{2}}$

Solution

$\begin{aligned} & \lim _{x \rightarrow 2}(x-1)^{\frac{1}{3 x-6}} \\ & =\lim _{x \rightarrow 2}(x-2+1)^{\frac{1}{3(x-2)}}=\lim _{x \rightarrow 2}\left\{[1+(x-2)]^{\frac{1}{(x-2)}}\right\}^{\frac{1}{3}}=e^{\frac{1}{3}}\end{aligned}$

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

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