함수의 극한과 연속 전체 문제

If $\lim_{x\to a}f(x)$ does not exist, then f is undefined at the point $x=a.$

Using the graph, find each limit or explain why the limit does not exist.

a. $\lim_{x\to -1}f(x)$ b. $\lim_{x\to 1}f(x)$ c. $\lim_{x\to {0}^{+}}f(x)$ d. $\lim_{x\to 2}f(x)$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\lim_{x\to 2}\frac{2{x}^{2}-3x-2}{x-2}$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\lim_{x\to 0}3{x}^{2}-2x+4$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\lim_{x\to 3}\frac{{x}^{3}-2{x}^{2}-1}{3x-2}$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\lim_{x\to \pi /2}\frac{\cot x}{\cos x}$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\lim_{x\to -5}\frac{{x}^{2}+25}{x+5}$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\lim_{x\to 2}\frac{3{x}^{2}-2x-8}{{x}^{2}-4}$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\lim_{x\to 1}\frac{{x}^{2}-1}{{x}^{3}-1}$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\lim_{x\to 1}\frac{{x}^{2}-1}{\sqrt{x}-1}$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\lim_{x\to 4}\frac{4-x}{\sqrt{x}-2}$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\lim_{x\to 4}\frac{1}{\sqrt{x}-2}$

In the following exercises, use the squeeze theorem to prove the limit.

$\lim_{x\to 0}{x}^{2}\cos (2\pi x)=0$

In the following exercises, use the squeeze theorem to prove the limit.

$\lim_{x\to 0}{x}^{3}\sin \left(\frac{\pi }{x}\right)=0$

Determine the domain such that the function $f(x)=\sqrt{x-2}+x{e}^{x}$ is continuous over its domain.

In the following exercises, determine the value of c such that the function remains continuous. Draw your resulting function to ensure it is continuous.

$f(x)=\left\{\begin{array}{l}{x}^{2}+1,x>c \\ 2x,x\le c\end{array}\right.$

In the following exercises, determine the value of c such that the function remains continuous. Draw your resulting function to ensure it is continuous.

$f(x)=\left\{\begin{array}{l}\sqrt{x+1},x>-1 \\ {x}^{2}+c,x\le -1\end{array}\right.$

In the following exercises, use the precise definition of limit to prove the limit.

$\lim_{x\to 1}(8x+16)=24$

In the following exercises, use the precise definition of limit to prove the limit.

$\lim_{x\to 0}{x}^{3}=0$

A ball is thrown into the air and the vertical position is given by $x(t)=-4.9{t}^{2}+25t+5.$ Use the Intermediate Value Theorem to show that the ball must land on the ground sometime between 5 sec and 6 sec after the throw.