Answer
$1$
Work Step by Step
Here, we have $\lim\limits_{t \to \infty} f(\infty)=\dfrac{\infty}{\infty}$
This shows an indeterminate form of limit, thus we will apply L-Hospital's rule such as:
$\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{p'(x)}{q'(x)}$
$\lim\limits_{t \to \infty} \dfrac{e^t+2t}{e^t-1}=\dfrac{e^{\infty}+2(\infty)}{e^{\infty}-1}=\dfrac{\infty}{\infty}$
Now, again apply L-Hospital's rule.
$\lim\limits_{t \to \infty} \dfrac{e^t}{e^t}=1$
