Limits and hyp functions

Question

Can someone please help me compute this ?

Lim (Cosh 2x) X->infinity

I could do this for tanh x as the "e"s raised to the negative powers will simple become zero...but i dont know what to do in this case

Help please.

Answer 1

$$ \cosh (2x) = \frac{e^{-2x} + e^{2x}}{2}$$ Can you take it from here?

Answer 2

$$\cosh(x) = \sum_{k=0}^\infty \frac{x^{2k}}{(2k)!} = 1+\frac{x^2}{2}+\frac{x^4}{24}+\frac{x^6}{720}+\cdots$$

It should be clear that all terms are positive (even if $x$ is negative), and that all terms except $1$ approach $\infty$ if $x$ does. So $$\lim_{x\to\pm\infty} \cosh(x) = +\infty$$