The limit of $((1+x)^{1/x}-e)/x\;$ as $\;x$ tends to $ 0$

Question

So, I have to solve the limit problem using a derivative.

The problem is as follows: $$\lim_{x\to0}\frac{(1+x)^{1/x}-e}{x}$$

I really don't know what to do. Can someone help?

Answer 1

Hint :

Observe that :

$$\lim_{x \to 0} (1+x)^{1/x} = \lim_{x\to 0} \left(1 + \frac{1}{\frac{1}{x}} \right)^{\frac{1}{x}} = e$$

Thus, the numerator of your fraction inside the limit tends to zero, while the same holds for the denominator.

Use L'Hopital's rule now and work on the calculation to derive the result.

Note : In case the entry line seemed weird to you, recall the sequential definition of $e$, as :

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