Chapter 13.1, Problem 17E

a.

To determine

To estimate: The value of the limit of the function $f(x)$ from the graph.

Answer to Problem 17E

The value of limit is $2$ .

Explanation of Solution

Given information:

  

The limit is $\underset{x\to 1^{-}}{{\displaystyle \lim }}f(x)$

Limit is $\underset{x\to 1^{-}}{{\displaystyle \lim }}f(x)$

From graph it can be observed that value of y − coordinate of x − coordinate which is slightly is less than $1$ is $2$

So, the value of $\underset{x\to 1^{-}}{{\displaystyle \lim }}f(x)$ is $2$

b.

To determine

To estimate: The value of the limit of function $f(x)$ from graph.

Answer to Problem 17E

The value of limit is $3$ .

Explanation of Solution

Given information:

Limit is $\underset{x\to 1^{+}}{{\displaystyle \lim }}f(x)$

  

The limit is $\underset{x\to 1^{+}}{{\displaystyle \lim }}f(x)$

From graph it can be observed that value of y − coordinate of x − coordinate which is slightly greater than $1$ is $3$ .

So, the value of $\underset{x\to 1^{+}}{{\displaystyle \lim }}f(x)$ is $3$

c.

To determine

To estimate: The value of the limit.

Answer to Problem 17E

The value of limit of the limit doesn’t exit.

Explanation of Solution

Given information:

The limit is $\underset{x\to 1}{{\displaystyle \lim }}f(x)$

  

Concept used:

When function is continuous and differentiable at point the limit exists.

Limit is $\underset{x\to 1}{{\displaystyle \lim }}f(x)$

  $\underset{x\to 1}{{\displaystyle \lim }}f(x)$ to be continuous it should follow

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

From above sub question a and b

  $\underset{x\to 1^{-}}{{\displaystyle \lim }}f(x)=2\underset{x\to 1^{+}}{{\displaystyle \lim }}f(x)=3$

So, $\underset{x\to 1^{-}}{{\displaystyle \lim }}f(x)\ne \underset{x\to 1^{+}}{{\displaystyle \lim }}f(x)$

Therefore limit doesn’t exist.

d.

To determine

To estimate: The value of the limit.

Answer to Problem 17E

The value of limit is $4$ .

Explanation of Solution

Given information:

Limit is $\underset{x\to 5}{{\displaystyle \lim }}f(x)$

  

Limit is $\underset{x\to 5}{{\displaystyle \lim }}f(x)$

From graph it can be observed at $x=5$ graph is continuous, that is

  $\underset{x\to 5}{{\displaystyle \lim }}f(x)=\underset{x\to 5^{+}}{{\displaystyle \lim }}f(x)=\underset{x\to 5}{{\displaystyle \lim }}f(x)=4$

e.

To determine

To estimate: The value of the function.

Answer to Problem 17E

The value of function is $4$ .

Explanation of Solution

Given information: $f\left(5\right)$

  

From graph it can be observed at $x=5$ graph is continuous, that is

  $\underset{x\to 5}{{\displaystyle \lim }}f(x)=\underset{x\to 5^{+}}{{\displaystyle \lim }}f(x)=\underset{x\to 5}{{\displaystyle \lim }}f(x)=4$

So, $f(5)=4$