$\lim _{x \rightarrow 1}\left(\lim _{7 \rightarrow \infty} y\left((e x)^{1 / y}-1\right)\right)=$

• A

  $\mathrm{e}$
• B

  0
• Correct answer
  C

  1
• D

  $-1$

Solution

$\begin{aligned} & =\lim _{x \rightarrow 1}\left(\lim _{y \rightarrow \infty} y\left(1+\frac{(x / y)}{1 !}+\frac{(x / y)^2}{2 !}+\frac{(x / y)^3}{3 !}+\ldots-1\right)\right) \\ & =\lim _{x \rightarrow 1}(x+0)=1+0=1 .\end{aligned}$

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