Chapter 11, Problem 20RE

a.

To determine

To know: The value of $\underset{x\to 3}{\lim }f\left(x\right)$ from the above graph

Answer to Problem 20RE

  $\underset{x\to 3}{\lim }f\left(x\right)=2$

Explanation of Solution

Given information:

Get the value of $f\left(x\right)$ at given x from the above graph.

Check the condition for existence of limit of a function at point condition of limit at point d.

  $\begin{array}{l}\underset{x\to d^{-}}{\lim }f\left(x\right)=\underset{x\to d^{+}}{\lim }f\left(x\right)=\underset{x\to d}{\lim }f\left(x\right)\\ =f\left(d\right)\end{array}$

From the above graph

  $\underset{x\to 3^{-}}{\lim }f\left(x\right)=2$

From the above graph

  $\underset{x\to 3^{+}}{\lim }f\left(x\right)=2$

  $\therefore$ Therefore limit exist at $x=2$

  $\underset{x\to 3}{\lim }f\left(x\right)=2$

b.

To determine

To know: The value of $\underset{x\to 1}{\lim }f\left(x\right)$ from the above graph

Answer to Problem 20RE

  $\underset{x\to 1}{\lim }f\left(x\right)=-1$

Explanation of Solution

Given information:

Get the value of $f\left(x\right)$ at given x from the above graph.

Check the condition for existence of limit of a function at point condition of limit at point d.

  $\begin{array}{l}\underset{x\to d^{-}}{\lim }f\left(x\right)=\underset{x\to d^{+}}{\lim }f\left(x\right)=\underset{x\to d}{\lim }f\left(x\right)\\ =f\left(d\right)\end{array}$

From the above graph

  $\underset{x\to 1}{\lim }f\left(x\right)=-1$

From the above graph

  $\underset{x\to 1^{+}}{\lim }f\left(x\right)=-1$

Hence $\underset{x\to 1}{\lim }f\left(x\right)=-1$

c.

To determine

To get: The value of $\underset{x\to -3}{\lim }f\left(x\right)$ from the above graph

Answer to Problem 20RE

  $\underset{x\to -3}{\lim }f\left(x\right)=1$

Explanation of Solution

Given information:

Get the value of $f\left(x\right)$ at given x from the above graph.

Check the condition for existence of limit of a function at point condition of limit at point d.

  $\underset{x\to d^{-}}{\lim }f\left(x\right)=\underset{x\to d^{+}}{\lim }f\left(x\right)=\underset{x\to d}{\lim }f\left(x\right)$

From the above graph

  $\underset{x\to -3^{-}}{\lim }f\left(x\right)=1$

From the above graph

  $\underset{x\to -3^{+}}{\lim }f\left(x\right)=1$

  $\therefore$ Therefore limit exists at $x=-3$

  $\underset{x\to -3}{\lim }f\left(x\right)=1$

d.

To determine

To get: The value of $\underset{x\to 2^{-}}{\lim }f\left(x\right)$ from the above graph

Answer to Problem 20RE

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

Explanation of Solution

Given information:

Extract the value of $f\left(x\right)$ at particular x from the graph.

From the above graph

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

Observe the value at $x\to 2^{-}$ and get the value.

e.

To determine

To get: The value of $\underset{x\to 2^{+}}{\lim }f\left(x\right)$ from the above graph

Answer to Problem 20RE

  $\underset{x\to 2^{+}}{\lim }f\left(x\right)=2$

Explanation of Solution

Given information:

Extract the value of $f\left(x\right)$ at particular x from the graph.

From the above graph

  $\underset{x\to 2^{+}}{\lim }f\left(x\right)=2$

Observe the value at $x\to 2^{+}$ and get the value.

f.

To determine

To get: The value of $f\left(x\right)$ from the given graph.

Answer to Problem 20RE

Limit does not exist.

Explanation of Solution

Given information:

Observe the value of $f\left(x\right)$ from the given graph.

Check the condition for existence of limit of a function at point condition of limit at point d.

  $\underset{x\to d^{-}}{\lim }f\left(x\right)=\underset{x\to d^{+}}{\lim }f\left(x\right)=\underset{x\to d}{\lim }f\left(x\right)$

From subparts d and e

  $\underset{x\to 2^{-}}{\lim }f\left(x\right)=1$ and $\underset{x\to 2^{+}}{\lim }f\left(x\right)=2$

  $\underset{x\to 2^{-}}{\lim }f\left(x\right)=1\ne \underset{x\to 2^{+}}{\lim }f\left(x\right)=2$

Then limit at $x=2$ does not exist.