4.3 Properties of Limits

In the previous section, we discussed what limits are by computing a function with a value that is approaching a certain number using a chart and a graph. This helped us understand the concept of limits. In this section, we are going to introduce some important properties of limits so that we do not have to graph or create a table to calculate the limit of a function.

As we discuss these properties, let $n$ be a real number, $c$ a constant, and $a$ a real number. Moreover, suppose that the following limits exist:

$\underset{x\to a}{lim}f(x)\text{and}\underset{x\to a}{lim}g(x)$

1. Limit of a Constant

First, the limit of any constant $c$ is the constant $c$:

$\underset{x\to a}{lim}c=c$

2. Limit of the Sum of Functions

The limit of a sum of functions is the sum of the limits of those functions.

$\underset{x\to a}{lim}[f(x)+g(x)]=\underset{x\to a}{lim}f(x)+\underset{x\to a}{lim}g(x)$

This rule applies to the subtraction as well.

$\underset{x\to a}{lim}[f(x)-g(x)]=\underset{x\to a}{lim}f(x)-\underset{x\to a}{lim}g(x)$

For example,

$\underset{x\to a}{lim}[x^5+2x^2]=\underset{x\to a}{lim}{x}^5+\underset{x\to a}{lim}2x^2$

3. Limit of the Product of a Constant and a Function

The limit of a constant times a function is the constant times the limit of the function.

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

For example, if we are finding the limit of $2x$ as $x$ approaches $a$,

$\underset{x\to a}{lim}2x=2\underset{x\to a}{lim}x$

4. Limit of the Product of Functions

The limit of a product of functions is the product of the limits of those functions.

$\underset{x\to a}{lim}[f(x)g(x)]=\underset{x\to a}{lim}f(x)\cdot \underset{x\to a}{lim}g(x)$

Suppose that $f(x)=3x$ and $g(x)=x^3+3x$. Then,

$\underset{x\to a}{lim}[3x(x^3+3x)]=\underset{x\to a}{lim}3x\cdot \underset{x\to a}{lim}(x^3+3x)$

5. Limit of the Quotient of Functions

The limit of a quotient of functions is the quotient of the limits of those functions.

$\underset{x\to a}{lim}\frac{f(x)}{g(x)}=\frac{\underset{x\to a}{lim}f(x)}{\underset{x\to a}{lim}g(x)}\text{if}\underset{x\to a}{lim}g(x)\ne 0$

Suppose that $f(x)=3x$ and $g(x)=x^3+3x$. Then,

$\underset{x\to a}{lim}\frac{3x}{x^3+3x}=\frac{\underset{x\to a}{lim}3x}{\underset{x\to a}{lim}(x^3+3x)}$

6. Limit of the Power of a Function

Given that the limit of a product of functions is the product of the limits of the functions, the limit of a power of a function is the power of the limit of the function.

$\underset{x\to a}{lim}[f(x){]}^n=[\underset{x\to a}{lim}f(x){]}^n$

This means that, for example, if $f(x)=x^{10}$, and we need to find the limit of the function as $x$ approaches $a,$ then

$\underset{x\to a}{lim}{x}^{10}=[\underset{x\to a}{lim}x{]}^{10}$

How about $f(x)=x^{-2}$?

$\underset{x\to a}{lim}{x}^{-2}=\frac{1}{(\underset{x\to a}{lim}x{)}^2}$

Finally, how about $f(x)=x^{\frac{2}{3}}$?

$\underset{x\to a}{lim}{x}^{\frac{2}{3}}=(\underset{x\to a}{lim}x{)}^{\frac{2}{3}}=\underset{x\to a}{lim}\sqrt[3]{x^2}=\sqrt[3]{\underset{x\to a}{lim}{x}^2}$

7. Limit of an Exponential Function

The limit of an exponential function is the limit of the exponent of the base number.

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

Suppose we want to find the limit of the exponential function $f(x)=e^{{\mathit{\text{x}}}^2+3\mathit{\text{x}}-4}$ with $x$ approaching $a$. Then,

$\underset{x\to a}{lim}{e}^{x^2+3x-4}=e^{\underset{x\to a}{lim}{x}^2+3x-4}$

8. Limit of a Logarithmic Function

Like the limit of an exponential function, the limit of a logarithmic function is the logarithm of the limit of the function.

$\underset{x\to a}{lim}[\ln f(x)]=\ln [\underset{x\to a}{lim}f(x)]$

For example, if $\underset{x\to a}{lim}[\ln f(x)]=\ln [\underset{x\to a}{lim}f(x)]$, and we need to find the limit of the function as $x$ approaches $a$,

$\underset{x\to a}{lim}[\ln (2x^5-3x-1)]=\ln [\underset{x\to a}{lim}(2x^5-3x-1)]$

9. Direct Substitution

If the function $f(x)$ is a polynomial or a rational function, and $a$ is in the domain of the function, then

$\underset{x\to a}{lim}[\ln f(x)]=f(a)$

For example, say that $f(x)=x^2$, and we want to find the limit of the function as $x$ approaches $a$. Since the domain of $x$ is all real numbers, which includes $a$, then

$\underset{x\to a}{lim}{x}^2=a^2$

Given this knowledge, what is the limit of the function $f(x)=x^3-10x+5$ as $x$ approaches 1?

$\underset{x\to 1}{lim}{x}^3-10x+5=1^3-10\cdot 1+5=-4$