극한의 성질과 계산

A function has to be continuous at $x=a$ if the $\lim_{x\to a}f(x)$ exists.

You can use the quotient rule to evaluate $\lim_{x\to 0}\frac{\sin x}{x}.$

If there is a vertical asymptote at $x=a$ for the function $f(x),$ then f is undefined at the point $x=a.$

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

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.

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

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 Exercises 7–14

, use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not.

• $\displaystyle \lim_{x\to9}f(x) = 6$, $\displaystyle \lim_{x\to6} f(x) = 9$, $f(9)=6$
• $\displaystyle \lim_{x\to9}g(x) = 3$, $\displaystyle \lim_{x\to6} g(x) = 3$, $g(6)=9$

$\displaystyle \lim_{x\to9}(f(x)+g(x))$

In Exercises 7–14

, use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not.

• $\displaystyle \lim_{x\to9}f(x) = 6$, $\displaystyle \lim_{x\to6} f(x) = 9$, $f(9)=6$
• $\displaystyle \lim_{x\to9}g(x) = 3$, $\displaystyle \lim_{x\to6} g(x) = 3$, $g(6)=9$

$\displaystyle \lim_{x\to9}(3f(x)/g(x))$

In Exercises 7–14

, use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not.

• $\displaystyle \lim_{x\to9}f(x) = 6$, $\displaystyle \lim_{x\to6} f(x) = 9$, $f(9)=6$
• $\displaystyle \lim_{x\to9}g(x) = 3$, $\displaystyle \lim_{x\to6} g(x) = 3$, $g(6)=9$

$\displaystyle \lim_{x\to9}\left(\frac{f(x)-2g(x)}{g(x)}\right)$