# Find the limits lim_{xrightarrow0}sec[e^x+pitan(frac{pi}{4sec x})-1]

Question
Limits and continuity
Find the limits
$$\lim_{x\rightarrow0}\sec[e^x+\pi\tan(\frac{\pi}{4\sec x})-1]$$

2021-02-22
Given
The given expression is $$\lim_{x\rightarrow0}\sec[e^x+\pi\tan(\frac{\pi}{4\sec x})-1]$$
find the limit
$$\lim_{x\rightarrow0}\sec[e^x+\pi\tan(\frac{\pi}{4\sec x})-1]=\sec[e^0+\pi\tan(\frac{\pi}{4\sec(0)})-1]$$
$$=\sec[1+\pi\tan(\frac{\pi}{4})-1]$$
$$=\sec(\pi)$$
$$=-1$$

### Relevant Questions

Use Taylor series to evaluate the following limits.
$$\lim_{x\rightarrow0}\frac{\sec x-\cos x-x^2}{x^4} \ (Hint: \text{The Maclaurin series for sec x is }1+\frac{x^2}{2}+\frac{5x^4}{24}+\frac{61x^6}{720}+...)$$
Compute the following limits if they exist $$\lim_{x\rightarrow0}\frac{\sec x-1}{x^3}$$
Find the limits
$$\lim_{x\rightarrow0}\sin(\frac{\pi+\tan x}{\tan x-2\sec x})$$
Use L'Hospital Rule to evaluate the following limits.
$$\lim_{x\rightarrow0}\frac{\tanh^{-1}x}{\tan(\pi x/2)}$$
Use Taylor series to evaluate the following limits. Express the result in terms of the nonzero real parameter(s).
$$\lim_{x\rightarrow0}\frac{e^{ax}-1}{x}$$
$$\lim_{x\rightarrow0}\frac{\frac{1}{x-1}+\frac{1}{x+1}}{x}$$
Suppose the functions f(x) and g(x) are defined for all x and that $$\lim_{x\rightarrow0}f(x)=\frac{1}{2}$$ and $$\lim_{x\rightarrow0}g(x)=\sqrt2$$. Find the limits as $$x\rightarrow0$$ of the following functions. $$f(x)\frac{\cos x}{x-1}$$
$$\lim_{x\rightarrow0}\frac{xe^x}{e^{3x}-1}$$
$$\lim_{x\rightarrow0}\frac{\sin x}{2x^2-x}$$
Find the limits. Write $$\infty$$ or $$-\infty$$ where appropriate. $$\lim_{x\rightarrow0^-}\frac{x^2-3x+2}{x^3-4x}$$