12.1 Finding Limits: Numerical and Graphical Approaches

For the following limit, define $a , f \left(x\right) ,$ and $L .$

For the following limit, define $a , f \left(x\right) ,$ and $L .$

Numerically estimate the following limit: $\underset{x \rightarrow 0}{\lim} \left(\sin \left(\frac{2}{x}\right)\right) .$

Explain the difference between a value at $x = a$ and the limit as $x$ approaches $a .$

$\underset{x \rightarrow - 2^{-}}{\lim} f \left(x\right)$

$\underset{x \rightarrow - 2}{\lim} f \left(x\right)$

$\underset{x \rightarrow 1^{-}}{\lim} f \left(x\right)$

$\underset{x \rightarrow 1}{\lim} f \left(x\right)$

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

$\underset{x \rightarrow 4}{\lim} f \left(x\right)$

Draw the graph of a function from the functional values and limits provided.

$\underset{x \rightarrow 0^{-}}{\lim} f \left(x\right) = 2$, $\underset{x \rightarrow 0^{+}}{\lim} f \left(x\right) = – 3$, $\underset{x \rightarrow 2}{\lim} f \left(x\right) = 2$, $f \left(0\right) = 4$, $f \left(2\right) = – 1$, $f \left(– 3\right) \textrm{ }\text{does}\textrm{ }\text{not}\textrm{ }\text{exist} .$

Draw the graph of a function from the functional values and limits provided.

$\underset{x \rightarrow 2^{-}}{\lim} f \left(x\right) = 0$, $\underset{x \rightarrow 2^{+}}{lim} = – 2$, $\underset{x \rightarrow 0}{\lim} f \left(x\right) = 3$, $f \left(2\right) = 5$, $f \left(0\right)$

Draw the graph of a function from the functional values and limits provided.

$\underset{x \rightarrow 2^{-}}{\lim} f \left(x\right) = 2$, $\underset{x \rightarrow 2^{+}}{\lim} f \left(x\right) = - 3$, $\underset{x \rightarrow 0}{\lim} f \left(x\right) = 5$, $f \left(0\right) = 1$, $f \left(1\right) = 0$

Draw the graph of a function from the functional values and limits provided.

$\underset{x \rightarrow 3^{-}}{\lim} f \left(x\right) = 0$, $\underset{x \rightarrow 3^{+}}{\lim} f \left(x\right) = 5$, $\underset{x \rightarrow 5}{\lim} f \left(x\right) = 0$, $f \left(5\right) = 4$, $f \left(3\right) \textrm{ }\text{does}\textrm{ }\text{not}\textrm{ }\text{exist} .$

Draw the graph of a function from the functional values and limits provided.

$\underset{x \rightarrow 4}{\lim} f \left(x\right) = 6$, $\underset{x \rightarrow 6^{+}}{\lim} f \left(x\right) = - 1$, $\underset{x \rightarrow 0}{\lim} f \left(x\right) = 5$, $f \left(4\right) = 6$, $f \left(2\right) = 6$

Draw the graph of a function from the functional values and limits provided.

$\underset{x \rightarrow - 3}{\lim} f \left(x\right) = 2$, $\underset{x \rightarrow 1^{+}}{\lim} f \left(x\right) = - 2$, $\underset{x \rightarrow 3}{\lim} f \left(x\right) = – 4$, $f \left(– 3\right) = 0$, $f \left(0\right) = 0$

Draw the graph of a function from the functional values and limits provided.

$\underset{x \rightarrow \pi}{\lim} f \left(x\right) = \pi^{2}$, $\underset{x \rightarrow – \pi}{\lim} f \left(x\right) = \frac{\pi}{2}$, $\underset{x \rightarrow 1^{–}}{\lim} f \left(x\right) = 0$, $f \left(\pi\right) = \sqrt{2}$, $f \left(0\right) \textrm{ }\text{does}\textrm{ }\text{not}\textrm{ }\text{exist} .$

Use a graphing calculator to determine the limit to 5 decimal places as $x$ approaches 0.

$g \left(x\right) = \left(1 + x\right)^{\frac{2}{x}}$

Use a graphing calculator to determine the limit to 5 decimal places as $x$ approaches 0.

$i \left(x\right) = \left(1 + x\right)^{\frac{4}{x}}$

Use a graphing calculator to determine the limit to 5 decimal places as $x$ approaches 0.

Based on the pattern you observed in the exercises above, make a conjecture as to the limit of $f \left(x\right) = \left(1 + x\right)^{\frac{6}{x}} ,$ $g \left(x\right) = \left(1 + x\right)^{\frac{7}{x}} ,$ $\text{and}\textrm{ } h \left(x\right) = \left(1 + x\right)^{\frac{n}{x}} .$

Use a graphing utility to find graphical evidence to determine the left- and right-hand limits of the function given as $x$ approaches $a .$ If the function has a limit as $x$ approaches $a ,$ state it. If not, discuss why there is no limit.

$\left(x\right) = \left\{\right. \begin{matrix}\frac{1}{x + 1} , & \text{if}\textrm{ } x = - 2 \\ \left(x + 1\right)^{2} , & \text{if}\textrm{ } x \neq - 2\end{matrix} a = - 2$

Use numerical evidence to determine whether the limit exists at $x = a .$ If not, describe the behavior of the graph of the function near $x = a .$ Round answers to two decimal places.

$f \left(x\right) = \frac{x^{2} - x - 6}{x^{2} - 9} ; a = 3$

Use numerical evidence to determine whether the limit exists at $x = a .$ If not, describe the behavior of the graph of the function near $x = a .$ Round answers to two decimal places.

$f \left(x\right) = \frac{x^{2} – 1}{x^{2} – 3 x + 2} ; a = 1$

Use numerical evidence to determine whether the limit exists at $x = a .$ If not, describe the behavior of the graph of the function near $x = a .$ Round answers to two decimal places.

$f \left(x\right) = \frac{10 - 10 x^{2}}{x^{2} - 3 x + 2} ; a = 1$

Use numerical evidence to determine whether the limit exists at $x = a .$ If not, describe the behavior of the graph of the function near $x = a .$ Round answers to two decimal places.

$f \left(x\right) = \frac{x}{4 x^{2} + 4 x + 1} ; a = - \frac{1}{2}$

Use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as $x$ approaches $a .$ If the function has a limit as $x$ approaches $a ,$ state it. If not, discuss why there is no limit.

$\underset{x \rightarrow 0}{\lim} e^{e^{- \frac{1}{x^{2}}}}$

Use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as $x$ approaches $a .$ If the function has a limit as $x$ approaches $a ,$ state it. If not, discuss why there is no limit.

$\underset{x \rightarrow - 1}{\lim} \frac{\left|x + 1\right|}{x + 1}$

Use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as $x$ approaches $a .$ If the function has a limit as $x$ approaches $a ,$ state it. If not, discuss why there is no limit.

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

Use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as $x$ approaches $a .$ If the function has a limit as $x$ approaches $a ,$ state it. If not, discuss why there is no limit.

$\underset{x \rightarrow 0}{\lim} \frac{5}{1 - e^{\frac{2}{x}}}$

According to the Theory of Relativity, the mass $m$ of a particle depends on its velocity $v$. That is

where $m_{o}$ is the mass when the particle is at rest and $c$ is the speed of light. Find the limit of the mass, $m ,$ as $v$ approaches $c^{-} .$