Evaluate the limit.
![\[ \lim_{x \to + \infty} \frac{\cosh (x+1)}{e^x}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-9d50ff7991b78e6531de8d8de9bea81d_l3.png "Rendered by QuickLaTeX.com")
We recall the definition of the hyperbolic cosine in terms of the exponential,
![\[ \cosh (x+1) = \frac{e^{x+1} + e^{-(x+1)}}{2}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-2f1bb52a94a04f0982e74933c5d2ae9b_l3.png "Rendered by QuickLaTeX.com")
Using this we compute,
![\begin{align*} \lim_{x \to +\infty} \frac{\cosh(x+1)}{e^x} &= \lim_{x \to +\infty} \frac{e^{x+1} + e^{-(x+1)}}{2e^x} \\[9pt] &= \lim_{x \to + \infty} \left( \frac{e}{2} + \frac{1}{2e^{2x+1}} \right) \\[9pt] &= \frac{e}{2}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-323cfc360849641d166c381f3b7e2787_l3.png "Rendered by QuickLaTeX.com")
Evaluate the limit.
We recall the definition of the hyperbolic cosine in terms of the exponential,
Using this we compute,