I need to evaluate the following limit, which is of the indeterminate form "$1^{\infty}$".
$$\lim_{x\to 0}\left(\frac{e}{(1+x)^{1/x}}\right)^{1/x}$$
The answer of this limit should be $\sqrt{e}$. I have tried it many times (using both binomial expansion and exponential expansion) but it is not working out. I have even tried online limit calculator like mathway but it says "I'm unable to solve this".
Then I tried MathPortal's Limit Calculator. This gives the answer correctly but doesn't shows steps. Then I tried WolframAlpha, but you can only access its step-by-step solution if you pro member (which is paid subscription) and I'm not one.
So if anybody can help me solve this limit it would be highly appreciative.
NOTE: I am a high school student.
Best Answer
Let $L$ be the required limit. Then, taking "logarithm":
$$\ln{L}=\lim_{x\to 0}\frac{\ln{e}-\ln((1+x)^{1/x})}{x}$$
$$=\lim_{x\to 0}\frac{1-\frac{\ln{(1+x)}}{x}}{x}$$
$$=\lim_{x\to 0}\left(\frac{x-\ln{(1+x)}}{x^2}\right)$$
Now try to find the above limit using L'Hospital Rule.