TITLE: Finding infinite limit of hyperbolic trig functions
QUESTION [6 upvotes]: I am trying to do random problems in my book and I do not know what to do for this one.
I am suppose to find
the limit as x approaches infinity of $\tanh x$
I really do not know what to do I know the problem is $ \frac{(e^x - e^{-x})/2}{(e^x + e^{-x})/2}$ which I think can reduce to $ \frac{e^x - e^{-x}}{e^x + e^{-x}}$.

REPLY [6 votes]: I'll give an intuitive idea:
$\sinh(x) = \frac{e^{x}-e^{-x}}{2}$
$\cosh(x) = \frac{e^{x}+e^{-x}}{2}$
Note that for (maybe even not so) large $x$, the $e^{-x}$ terms gets extremely small. So, $\tanh(x)$ "looks like"
$\tanh(x) = \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} \approx \frac{e^x}{e^x}=1$