calculus - Suppose that $lim a_n = 0$ Prove that $lim_{nrightarrow infty}left(1+a_nfrac {x}{n}right)^n = 1$.

Suppose that $\lim a_n = 0$ Prove that $$\lim_{n\rightarrow \infty}\left(1+a_n\frac {x}{n}\right)^n = 1$$

I was trying Leibniz theorem earlier but it was not working. Was I using the right one?

Answer

Using that $$\left(1+\frac{t}{n}\right)^n \to e^t=\exp(t)$$ as $n \to \infty$ with $a_nx:=t$ you obtain that $$\lim_{n\to \infty}\left(1+\frac{a_nx}{n}\right)^n=\lim_{n\to \infty} \exp{(a_nx)}=\ldots$$ Due to the continuity of the exponential function $f(t)=e^t$ you can interchange the $\lim$ operator with the $\exp$ operator to conclude that $$\ldots=\exp(\lim_{n\to \infty}a_nx)=\exp(0)=e^0=1$$