Chapter 1.8, Problem 66ES

Prove that ${\lim }_{x\to 0}{\left(1+x\right)}^{1/x}=e$ by completing parts (a) and (b).

(a) Use Equation $\left(4\right)$ and the substitution $t=1/x$ to prove that ${\lim }_{x\to 0^{+}}{\left(1+x\right)}^{1/x}=e$ .

(b) Use Equation $\left(5\right)$ and the substitution $t=1/x$ to prove that ${\lim }_{x\to 0^{+}}{\left(1+x\right)}^{1/x}=e$ .