Question

Hi, i need help with a calculus question involving limits. Could you explain how to solve this?

Answer 1

In order to solve this exercise, you have to know the limit that defines the constant $e$:

$\displaystyle \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n=e$

With this limit in mind, we can perform the following variable change:

$y=\dfrac{1}{x}$

With this new variable, we have

$x\to 0 \implies \frac{1}{x}=y\to\infty,\quad x = \dfrac{1}{y},\quad \dfrac{1}{x}=y$

Substituting these pieces one by one, we have

$\displaystyle \lim_{x\to 0}\left(1+x\right)^\frac{1}{x} = \lim_{y\to\infty}\left(1+\frac{1}{y}\right)^y$

and this is the limit that defines e.