How can I find a limit using equivalent functions and substitutions ( without applying L'Hospital) for the following problem:

Question

How can I find a limit using equivalent functions and substitutions ( without applying L'Hospital) for the following problem?

There you can see the expression, I need to find the limit for, as x tends to 0

$\exp[(\cos(\sqrt x)-1)/x]$, square brackets for clarity.

Answer 1

You may set $x = t^2$ and use

• $\lim_{y\to 0}\frac{1-\cos y}{y^2} = \frac{1}{2}$ (easy to verify with L'Hospital).

\begin{eqnarray*} e^{\frac{\cos\sqrt x -1}{x}} & = & e^{\frac{\cos t -1}{t^2}}\\ & \stackrel{t\to 0}{\longrightarrow} & e^{-\frac{1}{2}} =\frac{1}{\sqrt{e}}\\ \end{eqnarray*}