Chapter 13.1, Problem 19E

a.

To determine

To known: The value of $\underset{t\to 0^{-}}{\lim }g\left(t\right)$ from the above graph

Answer to Problem 19E

The value of $\underset{t\to 0^{-}}{\lim }g\left(t\right)$ is $-1$

Explanation of Solution

Given information: $\underset{t\to 0^{-}}{\lim }g\left(t\right)$

  

From the above graph

  $\underset{t\to 0^{-}}{\lim }g\left(t\right)=-1$

Observe graph at $t\to 0^{-}$ and get the value.

b.

To determine

To known: The value of $\underset{t\to 0^{+}}{\lim }g\left(t\right)$ from the above graph

Answer to Problem 19E

The value of $\underset{t\to 0^{+}}{\lim }g\left(t\right)$ is $-2$

Explanation of Solution

Given information: $\underset{t\to 0^{+}}{\lim }g\left(t\right)$

  

From the above graph

  $\underset{t\to 0^{+}}{\lim }g\left(t\right)=-2$

Observe graph at $t\to 0^{+}$ and get the value.

c.

To determine

To known: The value of $\underset{t\to 0}{\lim }g\left(t\right)$ from the above graph

Answer to Problem 19E

Non limit exists

Explanation of Solution

Given information: $\underset{t\to 0}{\lim }g\left(t\right)$

  

Check the condition for existence of limit of a function at given 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 subpart b and c

  $\underset{t\to 0^{-}}{\lim }g\left(t\right)=-1$ and $\underset{t\to 0^{+}}{\lim }g\left(t\right)=-2$

  $\underset{t\to 0^{-}}{\lim }g\left(t\right)\ne \underset{t\to 0^{+}}{\lim }g\left(t\right)$

Then limit at $t=0$ does not exist.

d.

To determine

To known: The value of $\underset{t\to 2^{-}}{\lim }g\left(t\right)$ from the above graph

Answer to Problem 19E

The value of $\underset{t\to 2^{-}}{\lim }g\left(t\right)$ is $2$

Explanation of Solution

Given information: $\underset{t\to 2^{-}}{\lim }g\left(t\right)$

  

From the above graph

  $\underset{t\to 2^{-}}{\lim }g\left(t\right)=2$

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

e.

To determine

To known: The value of $\underset{t\to 2^{+}}{\lim }g\left(t\right)$ from the above graph

Answer to Problem 19E

The value of $\underset{t\to 2^{+}}{\lim }g\left(t\right)$is $0$

Explanation of Solution

Given information: $\underset{t\to 2^{+}}{\lim }g\left(t\right)$

  

From the above graph

  $\underset{t\to 2^{+}}{\lim }g\left(t\right)=0$

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

f.

To determine

To known: The value of $\underset{t\to 2}{\lim }g\left(t\right)$ from the above graph

Answer to Problem 19E

  $\underset{t\to 2}{\lim }g\left(t\right)$ does not exist.

Explanation of Solution

Given information: $\underset{t\to 2}{\lim }g\left(t\right)$

  

Check the condition for existence of limit of a function at given 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 subpart b and c

  $\underset{t\to 2^{-}}{\lim }g\left(t\right)=2$ and $\underset{t\to 2^{+}}{\lim }g\left(t\right)=0$

  $\underset{t\to 2^{-}}{\lim }g\left(t\right)\ne \underset{t\to 2^{+}}{\lim }g\left(t\right)$

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

g.

To determine

To known: The value of $g\left(2\right)$ from the above graph

Answer to Problem 19E

The value of $g\left(2\right)$ is $1$

Explanation of Solution

Given information: $g\left(2\right)$

  

From above graph

  $g\left(2\right)=1$

Observe graph at $t=2$ and get the value of $g\left(2\right)$

h.

To determine

To get: The value of $\underset{t\to 4}{\lim }g\left(t\right)$ from the graph

Answer to Problem 19E

The value of $\underset{t\to 4}{\lim }g\left(t\right)$ is $3$

Explanation of Solution

Given information: $\underset{t\to 4}{\lim }g\left(t\right)$

  

Check the condition for existence of limit of a function at given 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 subpart b and c

  $\underset{t\to 4^{-}}{\lim }g\left(t\right)=3$ (from graph)

  $\underset{t\to 4^{+}}{\lim }g\left(t\right)=3$ (from graph)

Therefore $\underset{t\to 4}{\lim }g\left(t\right)=3$

  $\therefore$ Limit at $t=4$ exists.