One classical application of Taylor expansions is to obtain polynomial equivalents of complicated functions and use them to compute limits.
For example, with Landau notations, we have
$$\begin{array}{rcl}
\lim_{x\to 0}\frac{e^x-x-\cos(x)}{x^2} & = & \lim_{x\to 0}\frac{1+x+\frac{x^2}{2}-x-1+\frac{x^2}{2}+o(x^3)}{x^2} \\
& = & \lim_{x\to 0}\frac{x^2+o(x^2)}{x^2}\\
& = & \lim_{x\to 0} 1+o(1)=1
\end{array}
$$
But this example can be dealt with using L'Hospital's Rule twice. It seems to me that it would be always the case: since we basically consider ratio of "infinite degree polynomials", we can use repeatly l'Hospital's Rule in order to kill the indetermination.
My question: is there an example where Taylor expansions can be used but not L'Hospital's rule? I guess no so an example where computations with l'Hospital's Rule are awfully complicated but reasonable with Taylor would make me happy.
Best Answer
Yes, in principle you can always use l'Hopital's rule instead, but in practice there are a few reasons to prefer Taylor series expansions:
This point can be made more explicitly using functions with known Taylor series that are annoying to differentiate and then subtracting off several initial terms. For example, I personally wouldn't want to evaluate
$$\lim_{x \to 0} \frac{\tan x - x + \frac{x^3}{3}}{x^5}$$
by using l'Hopital's rule five times.