calculus - When a function contains a sequence, and how to find the function's limit?

Suppose $\lim \limits_{n \to \infty} a_n = 0$. Find the limit

$$\lim \limits_{n \to \infty} \left(1+a_n \frac{x}{n}\right)^n$$

It's kind intuitive that the answer is 1, but clearly I can't just say that the limits equal $\lim \limits_{n \to \infty} 1^n = 1$.

I feel like I should let $y =(1+a_n \frac{x}{n})^n$. Then take $log$ so that $log(y) = nlog(1+a_n \frac{x}{n})$ But L'hopital doesn't apply here either