Examples of limits that become easier with Taylor series

Question

There are examples of questions on this site where the OP asks for help solving a limit problem, and some of the answers make use of clever Taylor expansions to evaluate the limit. The purpose of this question is to collect examples of limit problems that become easier when a Taylor expansion is applied to the argument of the limit. For example, this question proposes the following limit $$ \lim_{x\to0}\left(\frac{1}{x^5}\int_0^xe^{-t^2}\,dt-\frac{1}{x^4}+\frac{1}{3x^2}\right) $$ that is readily solved by expanding $e^{-t^2} = 1 - t^2 + t^4/2 + O(t^6)$.

I am interested in other examples like this wherein a series expansion simplifies/makes possible the calculation of the limit. If the example also uses L'Hôpital's rule, that's fine. I am strictly interested in examples of expanding a function in a Taylor series to solve the limit.

Other examples here, here, and here.

Answer 1

Some nice examples :

$$\lim_{x\rightarrow 0} \frac{\sin(x)-x+\frac{x^3}{6}}{x^5}$$

$$\lim_{x\rightarrow 0} \frac{x\cdot \sqrt{x+1}-\ln({x+1})}{x^2}$$

$$\lim_{x\rightarrow 0} \frac{\ln{(x^2+1)}-\sin(x)\cdot \arctan(x)}{x^6}$$

$$\lim_{x\rightarrow 0} \frac{\exp(x)\cdot \cos(x)-\arctan(x)-1}{x^4}$$

$$\lim_{x\rightarrow 0} \frac{\sin{(\frac{x}{\exp(x)})}-\frac{x}{x+1}}{x^3}$$

$$\lim_{x\rightarrow 1} \frac{\sqrt{x}-\frac{1}{\sqrt{x}}-\ln{(x)}}{(x-1)^3}$$

Answer 2

Here are some limits which I personally consider to be especially suited to the technique of Taylor series expansions. Note that I have tried to solve most of them by using L'Hospital's Rule to show the difference between the use of Taylor series and L'Hospital's Rule.

1. $\displaystyle \lim_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$ Link.
2. $\displaystyle \lim_{x \to 0}\frac{\sin(\tan x) - \tan(\sin x)}{x^{7}}$ Link.
3. $\displaystyle \lim_{x \to 0}\frac{\sin^{2}x\tan x - x^{3}}{x^{7}}$ Link.
4. $\displaystyle \lim_{x \to 0}\frac{f(x) - 1}{x^{4}}$ where $\displaystyle f(x) = \dfrac{{\displaystyle 3\int_{0}^{x}(1 + \sec t)\log\sec t\,dt}}{(\log\sec x)\{x + \log(\sec x + \tan x)\}}$ Link.
5. $\displaystyle \lim_{x \to 0}\frac{(\cos x)^{\sin x} - \sqrt{1 - x^{3}}}{x^{6}}$ Link.
6. $\displaystyle \lim_{x \to 0}\frac{2\sin x \log \cos x + x^{3}}{x^{7}}$ Link.