Question

Evaluate :


$\displaystyle\lim_{x\to\zero}\: {(1+x )}^{\frac{1}{x}}$​

Answer 1

Answer:

Step-by-step explanation:

is this enough or do u want proof for the standard limit as well

Answer 2

ANSWER

$e^{1}$ or e.

Step By Step Explanation

$\displaystyle\lim_{x \rightarrow 0}\: {(1+x )}^{\frac{1}{x}}$

$(\rightarrow 1)^{\rightarrow\infty}$

$Since = \displaystyle\lim_{x\rightarrow a}\:{f(x)}^{g(x)} = e^{ \displaystyle\lim_{x\rightarrow a}\:(f(x)-1)g(x) }$

Now we have=>

$\implies e^{\displaystyle\lim_{x\rightarrow 0}(x + 1 - 1)\dfrac{1}{x} }$

$\implies e^{\displaystyle\lim_{x\rightarrow 0}(x \times \dfrac{1}{x}) }$

$\implies e^{\displaystyle\lim_{x\rightarrow 0}(1) }$

$\implies e^{1}$

Hence the evaluation of $\displaystyle\lim_{x\rightarrow 0}\: {(1+x )}^{\frac{1}{x}}$ is $\implies e^{1}$.