12.2 Finding Limits: Properties of Limits

Evaluate $\underset{x \rightarrow 3}{\lim} \left(2 x + 5\right) .$

Evaluate the following limit: $\underset{x \rightarrow - 12}{\lim} \left(- 2 x + 2\right) .$

Evaluate $\underset{x \rightarrow 3}{\lim} \left(5 x^{2}\right) .$

Evaluate $\underset{x \rightarrow 4}{\lim} \left(x^{3} - 5\right) .$

Evaluate $\underset{x \rightarrow 5}{\lim} \left(2 x^{3} - 3 x + 1\right) .$

Evaluate the following limit: $\underset{x \rightarrow - 1}{\lim} \left(x^{4} - 4 x^{3} + 5\right) .$

Evaluate $\underset{x \rightarrow 2}{\lim} \left(3 x + 1\right)^{5} .$

Evaluate the following limit: $\underset{x \rightarrow - 4}{\lim} \left(10 x + 36\right)^{3} .$

Evaluate $\underset{x \rightarrow 2}{\lim} \left(\frac{x^{2} - 6 x + 8}{x - 2}\right) .$

Evaluate the following limit: $\underset{x \rightarrow 7}{\lim} \left(\frac{x^{2} - 11 x + 28}{7 - x}\right) .$

Evaluate $\underset{x \rightarrow - 5}{\lim} \left(\frac{\frac{1}{5} + \frac{1}{x}}{10 + 2 x}\right) .$

Evaluate $\underset{x \rightarrow 0}{\lim} \left(\frac{\sqrt{25 - x} - 5}{x}\right) .$

Evaluate the following limit: $\underset{h \rightarrow 0}{\lim} \left(\frac{\sqrt{16 - h} - 4}{h}\right) .$

Evaluate $\underset{x \rightarrow 4}{\lim} \left(\frac{4 - x}{\sqrt{x} - 2}\right) .$

Evaluate the following limit: $\underset{x \rightarrow 3}{\lim} \left(\frac{x - 3}{\sqrt{x} - \sqrt{3}}\right) .$

Evaluate $\underset{x \rightarrow 7}{\lim} \frac{\left|x - 7\right|}{x - 7} .$

Evaluate $\underset{x \rightarrow 6^{+}}{\lim} \frac{6 - x}{\left|x - 6\right|} .$

Give an example of a type of function $f$ whose limit, as $x$ approaches $a ,$ is $f \left(a\right) .$

What does it mean to say the limit of $f \left(x\right) ,$ as $x$ approaches $c ,$ is undefined?

Evaluate the limits algebraically.

$\underset{x \rightarrow 2}{\lim} \left(\frac{- 5 x}{x^{2} - 1}\right)$

Evaluate the limits algebraically.

$\underset{x \rightarrow 3}{\lim} \left(\frac{x^{2} - 9}{x - 3}\right)$

Evaluate the limits algebraically.

$\underset{x \rightarrow \frac{3}{2}}{\lim} \left(\frac{6 x^{2} - 17 x + 12}{2 x - 3}\right)$

Evaluate the limits algebraically.

$\underset{x \rightarrow 3}{\lim} \left(\frac{x^{2} - 9}{x^{2} - 5 x + 6}\right)$

Evaluate the limits algebraically.

$\underset{x \rightarrow 3}{\lim} \left(\frac{x^{2} + 2 x - 3}{x - 3}\right)$

Evaluate the limits algebraically.

$\underset{h \rightarrow 0}{\lim} \left(\frac{\left(2 - h\right)^{3} - 8}{h}\right)$

Evaluate the limits algebraically.

$\underset{h \rightarrow 0}{\lim} \left(\frac{\sqrt{5 - h} - \sqrt{5}}{h}\right)$

Evaluate the limits algebraically.

$\underset{x \rightarrow 9}{\lim} \left(\frac{x^{2} - 81}{3 - \sqrt{x}}\right)$

Evaluate the limits algebraically.

$\underset{x \rightarrow 0}{\lim} \left(\frac{x}{\sqrt{1 + 2 x} - 1}\right)$

Evaluate the limits algebraically.

$\underset{x \rightarrow 4}{\lim} \left(\frac{x^{3} - 64}{x^{2} - 16}\right)$

Evaluate the limits algebraically.

$\underset{x \rightarrow 2^{+}}{\lim} \left(\frac{\left|x - 2\right|}{x - 2}\right)$

Evaluate the limits algebraically.

$\underset{x \rightarrow 4^{-}}{\lim} \left(\frac{\left|x - 4\right|}{4 - x}\right)$

Evaluate the limits algebraically.

$\underset{x \rightarrow 4}{\lim} \left(\frac{\left|x - 4\right|}{4 - x}\right)$

Use the given information to evaluate the limits: $\underset{x \rightarrow c}{\lim} f \left(x\right) = 3 ,$ $\underset{x \rightarrow c}{\lim} g \left(x\right) = 5$.

$\underset{x \rightarrow c}{\lim} \left[ 2 f \left(x\right) + \sqrt{g \left(x\right)} \right]$

Use the given information to evaluate the limits: $\underset{x \rightarrow c}{\lim} f \left(x\right) = 3 ,$ $\underset{x \rightarrow c}{\lim} g \left(x\right) = 5$.

$\underset{x \rightarrow c}{\lim} \frac{f \left(x\right)}{g \left(x\right)}$

Evaluate the following limits.

$\underset{x \rightarrow 2}{\lim} \sin \left(\pi x\right)$

Evaluate the following limits.

$f \left(x\right) = \left\{\right. \begin{matrix}2 x^{2} + 2 x + 1 , & x \leq 0 \\ x - 3 , & x > 0\end{matrix} ;\textrm{ } \underset{x \rightarrow 0^{+}}{\lim} f \left(x\right)$

Evaluate the following limits.

$f \left(x\right) = \left\{\right. \begin{matrix}2 x^{2} + 2 x + 1 , & x \leq 0 \\ x - 3 , & x > 0\end{matrix} ;\textrm{ } \underset{x \rightarrow 0}{\lim} f \left(x\right)$

Evaluate the following limits.

$\underset{x \rightarrow 2^{+}}{\lim} \left(2 x - \left[x\right]\right)$

Evaluate the following limits.

$\underset{x \rightarrow 3^{+}}{\lim} \frac{x^{2}}{x^{2} - 9}$

Find the average rate of change $\frac{f \left(x + h\right) - f \left(x\right)}{h} .$

$f \left(x\right) = 2 x^{2} - 1$

Find the average rate of change $\frac{f \left(x + h\right) - f \left(x\right)}{h} .$

$f \left(x\right) = x^{2} + 4 x - 100$

Find the average rate of change $\frac{f \left(x + h\right) - f \left(x\right)}{h} .$

$f \left(x\right) = \cos \left(x\right)$

Find the average rate of change $\frac{f \left(x + h\right) - f \left(x\right)}{h} .$

$f \left(x\right) = \frac{1}{x}$

Find the average rate of change $\frac{f \left(x + h\right) - f \left(x\right)}{h} .$

$f \left(x\right) = \sqrt{x}$

What is the left-hand limit of the function as $x$ approaches 0?

The height of a projectile is given by $s \left(t\right) = - 64 t^{2} + 192 t$ Find the average rate of change of the height from $t = 1$ second to $t = 1.5$ seconds.