# Solve this limit $\lim_{x\to0} (1+x)^{\frac{1}{x}}$ [closed]

How would I go about solving this limit ?

$$\lim_{x\to0} \,(1+x)^{\frac{1}{x}}$$

I tried several times to solve it but I can't...

## closed as off-topic by Travis, Davide Giraudo, drhab, Shaun, voldemortJan 23 '15 at 13:22

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• Hint: Write $x = \frac{1}{n}$. – Travis Jan 23 '15 at 11:31
• think of the limit of $(1+\frac{1}{n})^n$ when $n\rightarrow \infty$. – Ofir Schnabel Jan 23 '15 at 11:31
• Rather than saying "I tried several times to solve it but I can't..." which doesn't add anything, you could share what exactly you have tried so far. – Hippalectryon Jan 23 '15 at 11:57

Hint:

Using the binomial theorem with $x=1/n$, you have

$$(1+\frac1n)^n=1+\frac nn+\frac {n(n-1)}{2n^2}+\frac{n(n-1)(n-2)}{3!n^3}+\cdots\le1+\frac11+\frac12+\frac1{3!}\cdots.$$ The sum is bounded and every term converges to the corresponding one in the last expression, that we can denote $e$.

• Nice explanation – Vim Jan 23 '15 at 12:31

Hint

Substitute $t = 1/x$ and remember a famous limit

Hint: rewrite this as $\exp(\frac{\ln(1+x)}{x})$ as apply l'Hopital.

• using l'Hoptial's rule depends on the derivative of exp/log, which in some constructions depends on the value of this very limit. So this could be a circular argument. – Matthew Leingang Jan 23 '15 at 11:36