What is this limit?

Question

Evaluate $\displaystyle\lim_{x \to 0} 2 \frac {(\cosh x-1)^{1/x^2}}{x^2} $.

After manipulating I got it equal to $\lim\limits_{x \to 0} 2 \dfrac {(e^{\frac{x}{2}}-e^{\frac {-x}{2}})^{\frac {2} {x^2}}} {x^2}$. What now? Should I take log?

Answer 1

The factor $2$ is completely irrelevant, so we'll put it back at the end.

You can observe that the function is even, so computing the limit for $x\to0^+$ is sufficient. In this case we can do the substitution $x=\sqrt{t}$ and transform it into $$ \lim_{t\to0^+}\frac{(\cosh\sqrt{t}-1)^{1/t}}{t} $$ The idea of taking the logarithm is very good, because we need to compute $$ \lim_{t\to0^+}\frac{\log(\cosh\sqrt{t}-1)}{t}-\log t= \lim_{t\to0^+}\frac{\log(\cosh\sqrt{t}-1)-t\log t}{t} $$ which should be easier to compute: remember that $\lim_{t\to0^+}t\log t=0$.