Chapter 1, Problem 1RE

For the function $f$ graphed in the accompanying figure, find the limit if it exists.

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

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

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

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

(e) $\underset{x\to +\infty }{\lim }f\left(x\right)$

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

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

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

(i) $\underset{x\to 0}{\lim }f\left(x\right)$

To determine

The value of $\underset{x\to 1}{\lim }f\left(x\right)$ if it exists.

Answer to Problem 1RE

The value of $\underset{x\to 1}{\lim }f\left(x\right)$ exists and is equal to 1.

Explanation of Solution

Consider the graph.

It is clear from the graph that the function is continuous at $x=1$ and its value is 1.

This implies,

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

Hence, the value of $\underset{x\to 1}{\lim }f\left(x\right)$ exists and is equal to 1.

To determine

The value of $\underset{x\to 2}{\lim }f\left(x\right)$ if it exists.

Answer to Problem 1RE

The value of $\underset{x\to 2}{\lim }f\left(x\right)$ does not exist.

Explanation of Solution

Consider the figure.

It is clear from the graph that the function approaches to 2 as $x$ tends to 2 from the left and approaches to 1 as $x$ tends to 2 from the right.

This implies that the left hand limit is not equal to right hand limit.

Therefore, $\underset{x\to 2}{\lim }f\left(x\right)$ does not exist.

To determine

The value of $\underset{x\to 3}{\lim }f\left(x\right)$ if it exists.

Answer to Problem 1RE

The value of $\underset{x\to 3}{\lim }f\left(x\right)$ does not exist.

Explanation of Solution

Consider the given figure.

It is clear from the graph that the function approaches to 2 as $x$ tends to 3 from the left and approaches to 0 as $x$ tends to 3 from the right.

Hence, $\underset{x\to 3}{\lim }f\left(x\right)$ does not exist.

To determine

The value of $\underset{x\to 4}{\lim }f\left(x\right)$ if it exists.

Answer to Problem 1RE

The value of $\underset{x\to 4}{\lim }f\left(x\right)$ exists and is equal to 1.

Explanation of Solution

Consider the given figure.

It can be seen from the graph that the function $f\left(x\right)$ approaches to 1 as $x$ tends to 4 either from the left or from the right.

This implies,

$\underset{x\to 4}{\lim }f\left(x\right)=\text{1}$

Hence, the value of $\underset{x\to 4}{\lim }f\left(x\right)$ exists and is equal to 1.

To determine

The value of $\underset{x\to +\infty }{\lim }f\left(x\right)$ if it exists.

Answer to Problem 1RE

The value of $\underset{x\to +\infty }{\lim }f\left(x\right)$ exists and is equal to 3.

Explanation of Solution

Consider the figure.

It can be observed from the graph that the function $f\left(x\right)$ can attain maximum value 3. Thus, for sufficiently large $x$ , the function approaches to 3.

This implies,

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

Hence, the value of $\underset{x\to +\infty }{\lim }f\left(x\right)$ exists and is equal to 3.

To determine

The value of $\underset{x\to -\infty }{\lim }f\left(x\right)$ if it exists.

Answer to Problem 1RE

The value of $\underset{x\to -\infty }{\lim }f\left(x\right)$ exists and is equal to 0.

Explanation of Solution

Consider the figure.

It can be observed from the graph that the minimum value that function can approach to is 0. Therefore, the function approaches to 0 for sufficiently small $x$ .

This implies,

$\underset{x\to -\infty }{\lim }f\left(x\right)=0$

Hence, the value of $\underset{x\to -\infty }{\lim }f\left(x\right)$ exists and is equal to 0.

To determine

The value of $\underset{x\to 3^{+}}{\lim }f\left(x\right)$ if it exists.

Answer to Problem 1RE

The value of $\underset{x\to 3^{+}}{\lim }f\left(x\right)$ exists and is equal to 0.

Explanation of Solution

Consider the figure.

It can be clearly seen from the graph, the function approaches to 0 as $x$ tends to 3 from the right.

This implies,

$\underset{x\to 3^{+}}{\lim }f\left(x\right)=0$

Hence, the value of $\underset{x\to 3^{+}}{\lim }f\left(x\right)$ exists and is equal to 0.

To determine

The value of $\underset{x\to 3^{-}}{\lim }f\left(x\right)$ if it exists.

Answer to Problem 1RE

The value of $\underset{x\to 3^{-}}{\lim }f\left(x\right)$ exists and is equal to 2.

Explanation of Solution

Consider the figure.

It can be observed from the graph that the function approaches to 2 as $x$ tends to 3 from the left.

This implies,

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

Hence, the value of $\underset{x\to 3^{-}}{\lim }f\left(x\right)$ exists and is equal to 2.

To determine

The value of $\underset{x\to 0}{\lim }f\left(x\right)$ if it exists.

Answer to Problem 1RE

The value of $\underset{x\to 0}{\lim }f\left(x\right)$ exists and is equal to $\frac{1}{2}$ .

Explanation of Solution

Consider the given figure.

It can be seen from the graph that the function $f\left(x\right)$ attains value in the interval $\left(0,1\right)$ at $x=0$ . Thus, it can be concluded that the function approximately approaches to $\frac{1}{2}$ as $x$ tends to 0.

This implies,

$\underset{x\to 0}{\lim }f\left(x\right)=\frac{1}{2}$

Hence, the value of $\underset{x\to 0}{\lim }f\left(x\right)$ exists and is equal to $\frac{1}{2}$ .