How is $\lim_{x\to 0}[(1+5x)^{\frac{1}{5x}}]^{10}=e^{10}$?

Question

Problem:

I was doing trying to find the value of $$\lim_{x\to 0}(1+5x)^{\frac{3x+2}{x}}$$

and I at some point reached these steps in my book:

$$\lim_{x\to 0}(1+5x)^3.\lim_{x\to 0}[(1+5x)^{\frac{1}{5x}}]^{10}$$

$$(1+5\times0)^3.e^{10}$$

Question:

1. How does $\lim_{x\to 0}[(1+5x)^{\frac{1}{5x}}]^{10}=e^{10}$?

Answer 1

It is based on the definition of $e$: \begin{align*} e = \exp(1) = \lim_{x\to 0}(1 + x)^{1/x} \end{align*}

So you can obtain the desired result by means of the substitution $x = 5y$.