Can you think of a simple example of computing $\lim_{x\to a}f(x)$ by directly evaluating $f(x)$ on a sequence that converges to $a$ and passing to a limit? I'm searching for a motivating example for students that do not specialize in math.
Consider for instance, $f(x)=x\ln x$. You can evaluate it on $\{\frac1n\}$, but this would result in $-\frac{\ln n}{n}$, which does not bring us any further. Let's consider $\{e^{-n}\}$. Again, we have $-ne^{-n}$, which is not obvious.
I'd appreciate an example, where the solution is obvious, that is one does not need to employ comparison theorems for computing the limit $\lim_{n\to\infty} f(y_n)$.