Textbook Problem

Explain what each of the following means and illustrate with sketch.

(a) $\underset{x\to a}{\lim }f(x)=L$

(b) $\underset{x\to a^{+}}{\lim }f(x)=L$

(c) $\underset{x\to a^{-}}{\lim }f(x)=L$

(d) $\underset{x\to a}{\lim }f(x)=\infty$

(e) $\underset{x\to \infty }{\lim }f(x)=L$

To determine

To explain: The meaning of $\underset{x\to a}{\lim }f\left(x\right)=L$.

Explanation of Solution

Result used:

Definition of limit:

Let $f\left(x\right)$ be a function is defined when $x$ approaches to $p$ then $\underset{x\to p}{\lim }f\left(x\right)=L$, if for every number $\epsilon >0$ there is some $\delta >0$  such that $\left|f\left(x\right)-L\right|<\epsilon$ whenever $0<\left|x-p\right|<\delta$.

Graph:

Calculation:

The limit of the function $\underset{x\to a}{\lim }f\left(x\right)=L$ means the limit of $f\left(x\right)$ equal to $L$ when $x$ approaches to $a$, if $x$ is closer and closer to $a$ from the both sides then the value of $f\left(x\right)$ also closer and closer to $L$.

In the limit definition $x\ne a$ this means finding the limit of $f\left(x\right)$ when $x$ approaches to $a$, there no need to consider $x=a$.

There are three cases for define $\underset{x\to a}{\lim }f\left(x\right)=L$.

Case (1):

The limit of the function $\underset{x\to a}{\lim }f\left(x\right)=L$, if $x$ approaches to $a$ then the value of $f\left(x\right)$ are closer to $L$ and $f\left(a\right)$ is $L$.

Graph:

Case (2):

The limit of the function $\underset{x\to a}{\lim }f\left(x\right)=L$, if $x$ approaches to $a$ then the value of $f\left(x\right)$ are closer to $L$ and $f\left(a\right)$ is undefined.

Case (3):

The limit of the function $\underset{x\to a}{\lim }f\left(x\right)=L$, if $x$ approaches to $a$ then the value of $f\left(x\right)$ are closer to other than $L$.

Graph:

To determine

To explain: The meaning of $\underset{x\to a^{+}}{\lim }f\left(x\right)=L$.

Explanation of Solution

Result used:

Definition of limit:

Let $f\left(x\right)$ be a function is defined when $x$ approaches to $p$ then $\underset{x\to p}{\lim }f\left(x\right)=L$, if for every number $\epsilon >0$ there is some $\delta >0$  such that $\left|f\left(x\right)-L\right|<\epsilon$ whenever $0<\left|x-p\right|<\delta$.

Calculation:

$\underset{x\to a^{+}}{\lim }f\left(x\right)=L$ means the limit of $f\left(x\right)$ equal to $L$ when $x$ approaches to $a$ from the right, if $x$ is closer and closer to $a$ from the right and remains greater than a then the value of $f\left(x\right)$ also closer and closer to $L$.

Graph:

To determine

To explain: The meaning of $\underset{x\to a^{-}}{\lim }f\left(x\right)=L$.

Explanation of Solution

Result used:

Definition of limit:

Let $f\left(x\right)$ be a function is defined when $x$ approaches to $p$ then $\underset{x\to p}{\lim }f\left(x\right)=L$, if for every number $\epsilon >0$ there is some $\delta >0$  such that $\left|f\left(x\right)-L\right|<\epsilon$ whenever $0<\left|x-p\right|<\delta$.

Calculation:

The limit of the function $\underset{x\to a^{-}}{\lim }f\left(x\right)=L$ means the limit of $f\left(x\right)$ equal to $L$ when $x$ approaches to $a$ from the left, if $x$ is closer and closer to $a$ from the left and remains less than a then the value of $f\left(x\right)$ also closer and closer to $L$.

Graph:

To determine

To explain: The meaning of $\underset{x\to a}{\lim }f\left(x\right)=\infty$.

Explanation of Solution

Result used:

Definition of limit:

Let $f\left(x\right)$ be a function is defined when $x$ approaches to $p$ then $\underset{x\to p}{\lim }f\left(x\right)=L$, if for every number $\epsilon >0$ there is some $\delta >0$  such that $\left|f\left(x\right)-L\right|<\epsilon$ whenever $0<\left|x-p\right|<\delta$.

Calculation:

The limit of the function $\underset{x\to a}{\lim }f\left(x\right)=\infty$ means the limit of $f\left(x\right)$ is larger value when $x$ approaches to $a$ from the both sides. That is any $M>0$, $f\left(x\right)>M$ for some x-value is sufficiently close to a.

Graph:

To determine

To explain: The meaning of $\underset{x\to \infty }{\lim }f\left(x\right)=L$.

Explanation of Solution

Result used:

Definition of limit:

Let $f\left(x\right)$ be a function is defined when $x$ approaches to $p$ then $\underset{x\to p}{\lim }f\left(x\right)=L$, if for every number $\epsilon >0$ there is some $\delta >0$  such that $\left|f\left(x\right)-L\right|<\epsilon$ whenever $0<\left|x-p\right|<\delta$.

Calculation:

The limit of the function $\underset{x\to \infty }{\lim }f\left(x\right)=L$ means the limit of $f\left(x\right)$ is L when $x$ approaches to larger value the graph get closer and closer to the line $y=L$.

Graph: