Question

For the function f whose graph is given, state the value of the given quantity if it exists. If it does not exist, explain why.

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

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

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

$\left(d\right)\underset{x\to 5}{\lim }f\left(x\right)$

$\left(e\right)f\left(5\right)$

Short answer

(a)

The value of limit of the given function is$2.$

Step-by-step solution

Step 1. Given information.

The graph of the limits are given here,

Step 2. Formula used.

Here, we use the concept of limits from a graph.

Step 3. Simplification.

Limit of f as x approaches$1^{-}$ from the left side is$2$

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

Therefore, our final answer is2.

(b)

The value of limit of the given function is$3.$

Step 1. Given information.

The graph of the limits are given here,

Step 2. Formula used.

Here, we use the concept of limits from a graph.

Step 3. Simplification.

Limit of f as x approaches $1$ from the right side is $3.$

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

Therefore, our final answer is $3.$

(c)

The value of limit of the given function is

Step 1. Given information.

The graph of the limits are given here,

Step 2. Formula used.

Here, we use the concept of limits from a graph.

Step 3. Simplification.

Since the left- and right-hand limits are different, we conclude that $\underset{x\to 1}{\lim }f\left(x\right)$

does not exist.

Therefore, our final answer is $does\text{notexist}.$

(d)

The value of limit of the given function is$4.$

Step 1. Given information.

The graph of the limits are given here,

Step 2. Formula used.

Here, we use the concept of limits from a graph.

Step 3. Simplification.

Limit of f as x approaches $5$ from the right side is $4$

$\underset{x\to 5}{\lim }f\left(x\right)=4$

Therefore, our final answer is $4.$

(e)

The value of limit of the given function is$not\text{defined}.$

Step 1. Given information.

The graph of the limits are given here,

Step 2. Formula used.

Here, we use the concept of limits from a graph.

Step 3. Simplification.

Limit of f as x approaches can’t be done.

As,

$\underset{x\to 5}{\lim }f\left(x\right)=4$

So,$f5$can’t define.

Therefore, our final answer is$\text{notdefined}.$