IB

# The Limit of a Function and One Sided Limits

Essentially, for us to say the limit, L, of some function, f, actually exists as x approaches some value, say c, then we must have it that both the one sided limits must be present and equal the same value, L, otherwise we say the the limit of f as x approaches...

#### Definition of Limit

The Statement

$\underset{x\to c}{\mathrm{lim}}f\left(x\right)=L$

means  f  approaches the limit  L  as  x  approaches  c.

Which is read  “the limit of  f(x), as  x  approaches  c, equals L.”

#### Basic Limits

1. If  f  is the constant function f(x) = k, then for any value of c,

$\underset{x\to c}{\mathrm{lim}}f\left(x\right)=\underset{x\to c}{\mathrm{lim}}k=k$

2. If  f  is the polynomial function  f(x) = xn, then for any value of c,

$\underset{x\to c}{\mathrm{lim}}f\left(x\right)=\underset{x\to c}{\mathrm{lim}}{x}^{n}={c}^{n}$

#### Finding Limits Graphically

Consider the graph of the function

$f\left(x\right)=\frac{{x}^{3}+1}{x+1}$

The given function is defined for all real numbers x except x = −1. The graph of f is a parabola with the point (−1, 3) removed as shown below. Even though f(−1) is not defined, we can make the value of f(x) as close to 3 as we want by choosing an x close enough to −1.

Although  f(x)  is not defined when  x=−1, the limit of  f(x)  as x approaches  −1 is 3, because the definition of a limit says that we consider values of x that are close to c, but not equal to c.

#### One Sided Limits

The right-hand Limit means that x approaches c from values greater than c.

This limit is denoted as

$\underset{x\to {c}^{+}}{\mathrm{lim}}f\left(x\right)=L$

The left-hand limit means that x approaches c from values less than c.

This limit is denoted as

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

#### The Existence of a Limit

The limit of f(x) as x approaches c is L if and only if

$\underset{x\to {c}^{+}}{\mathrm{lim}}f\left(x\right)=L$

and

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

#### Limits That Fail to Exits

Some limits that fail to exits are illustrated below.

The left limit and the right limit is different.

$\begin{array}{l}\underset{x\to 0}{\mathrm{lim}}\frac{\left|x\right|}{x}=1,\text{\hspace{0.17em}}if\text{\hspace{0.17em}}x>0\text{\hspace{0.17em}}and\\ \underset{x\to 0}{\mathrm{lim}}\frac{\left|x\right|}{x}=-1,\text{\hspace{0.17em}}if\text{\hspace{0.17em}}x<0\end{array}$

As x approaches 0 from  from the right or the left,  f(x)  increases or decreases without bound.

The values of  f(x)  oscillate between −1 and 1 infinitely often as  x  approaches 0.

Example 1. Find the limit.

$\begin{array}{l}\left(a\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to 2}{\mathrm{lim}}\left(7\right)\hfill \\ \left(b\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to -1}{\mathrm{lim}}\left({x}^{3}-2x\right)\hfill \\ \left(c\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}x}{x}\hfill \end{array}$

Solution:

$\begin{array}{l}\left(a\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to 2}{\mathrm{lim}}\left(7\right)=7\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}because\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to c}{\mathrm{lim}}k=k\\ \left(b\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to -1}{\mathrm{lim}}\left({x}^{3}-2x\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}={\left(}^{-}-2\left(-1\right)=1\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}because\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to c}{\mathrm{lim}}{x}^{n}={c}^{n}\end{array}$

(c) The function f(x)  is not defined when x = 0. Find the Limit Graphically.

$\text{Graph}\text{\hspace{0.17em}}y=\frac{\mathrm{sin}x}{x}$

using a graphing calculator. The limit of f(x) = (sin x)/x as x approaches 0 is 1.

Example 2. The Graph of the function  f  is shown in the figure below. Find the limit or value of the function at a given point.

$\begin{array}{l}\left(a\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to {3}^{-}}{\mathrm{lim}}f\left(x\right)\\ \left(b\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to {3}^{+}}{\mathrm{lim}}f\left(x\right)\\ \left(c\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to 3}{\mathrm{lim}}f\left(x\right)\\ \left(d\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to 6}{\mathrm{lim}}f\left(x\right)\\ \left(e\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(3\right)\\ \left(f\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(6\right)\end{array}$

Solution:

$\begin{array}{l}\left(a\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to {3}^{-}}{\mathrm{lim}}f\left(x\right)=0\\ \left(b\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to {3}^{+}}{\mathrm{lim}}f\left(x\right)=3\\ \left(c\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to 3}{\mathrm{lim}}f\left(x\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{does}\text{\hspace{0.17em}}\text{not}\text{\hspace{0.17em}}\text{exists}\text{\hspace{0.17em}}\text{since}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to {3}^{-}}{\mathrm{lim}}f\left(x\right)\ne \underset{x\to {3}^{+}}{\mathrm{lim}}f\left(x\right)\\ \left(d\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to 6}{\mathrm{lim}}f\left(x\right)=3,\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}because\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to {6}^{-}}{\mathrm{lim}}f\left(x\right)=3=\underset{x\to {6}^{+}}{\mathrm{lim}}f\left(x\right)\\ \left(e\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(3\right)=0\\ \left(f\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(6\right)=1\end{array}$

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