Answer
$1$
Work Step by Step
L'Hospital's rule states that $\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{A'(x)}{B'(x)}$
Here, $\lim\limits_{t \to \infty} f(\infty)=\dfrac{\infty}{\infty}$
This shows an indeterminate form of the limit, so we need to use L'Hospital's rule.
$\lim\limits_{t \to \infty} \dfrac{e^t+2t}{e^t-1}=\dfrac{\infty}{\infty}$
This shows an indeterminate form of the limit, so we need to use L'Hospital's rule.
$\lim\limits_{t \to \infty} \dfrac{e^t}{e^t}=\dfrac{e^0}{e^0}=1$