Evaluating Limits

Learning Outcomes

  • Recognize the basic limit laws
  • Use the limit laws to evaluate the limit of a function

Limit Laws

The first two limit laws were stated earlier in the course and we repeat them here. These basic results, together with the other limit laws, allow us to evaluate limits of many algebraic functions.

Basic Limit Results

For any real number [latex]a[/latex] and any constant [latex]c[/latex],

  1. [latex]\underset{x\to a}{\lim}x=a[/latex]
  2. [latex]\underset{x\to a}{\lim}c=c[/latex]

Example: Evaluating a Basic Limit

Evaluate each of the following limits using the basic limit results above.

  1. [latex]\underset{x\to 2}{\lim}x[/latex]
  2. [latex]\underset{x\to 2}{\lim}5[/latex]

Try It

We now take a look at the limit laws, the individual properties of limits. The proofs that these laws hold are omitted here.

Limit Laws

Let [latex]f(x)[/latex] and [latex]g(x)[/latex] be defined for all [latex]x\ne a[/latex] over some open interval containing [latex]a[/latex]. Assume that [latex]L[/latex] and [latex]M[/latex] are real numbers such that [latex]\underset{x\to a}{\lim}f(x)=L[/latex] and [latex]\underset{x\to a}{\lim}g(x)=M[/latex]. Let [latex]c[/latex] be a constant. Then, each of the following statements holds:

 

Sum law for limits: [latex]\underset{x\to a}{\lim}(f(x)+g(x))=\underset{x\to a}{\lim}f(x)+\underset{x\to a}{\lim}g(x)=L+M[/latex]

 

Difference law for limits: [latex]\underset{x\to a}{\lim}(f(x)-g(x))=\underset{x\to a}{\lim}f(x)-\underset{x\to a}{\lim}g(x)=L-M[/latex]

 

Constant multiple law for limits: [latex]\underset{x\to a}{\lim}cf(x)=c \cdot \underset{x\to a}{\lim}f(x)=cL[/latex]

 

Product law for limits: [latex]\underset{x\to a}{\lim}(f(x) \cdot g(x))=\underset{x\to a}{\lim}f(x) \cdot \underset{x\to a}{\lim}g(x)=L \cdot M[/latex]

 

Quotient law for limits: [latex]\underset{x\to a}{\lim}\dfrac{f(x)}{g(x)}=\dfrac{\underset{x\to a}{\lim}f(x)}{\underset{x\to a}{\lim}g(x)}=\frac{L}{M}[/latex] for [latex]M\ne 0[/latex]

 

Power law for limits: [latex]\underset{x\to a}{\lim}(f(x))^n=(\underset{x\to a}{\lim}f(x))^n=L^n[/latex] for every positive integer [latex]n[/latex].

 

Root law for limits: [latex]\underset{x\to a}{\lim}\sqrt[n]{f(x)}=\sqrt[n]{\underset{x\to a}{\lim}f(x)}=\sqrt[n]{L}[/latex] for all [latex]L[/latex] if [latex]n[/latex] is odd and for [latex]L\ge 0[/latex] if [latex]n[/latex] is even

We now practice applying these limit laws to evaluate a limit.

Example: Evaluating a Limit Using Limit Laws

Use the limit laws to evaluate [latex]\underset{x\to -3}{\lim}(4x+2)[/latex].

Example: Using Limit Laws Repeatedly

Use the limit laws to evaluate [latex]\underset{x\to 2}{\lim}\dfrac{2x^2-3x+1}{x^3+4}[/latex].

Try It

Use the limit laws to evaluate [latex]\underset{x\to 6}{\lim}(2x-1)\sqrt{x+4}[/latex]. In each step, indicate the limit law applied.

Limits of Polynomial and Rational Functions

By now you have probably noticed that, in each of the previous examples, it has been the case that [latex]\underset{x\to a}{\lim}f(x)=f(a)[/latex]. This is not always true, but it does hold for all polynomials for any choice of [latex]a[/latex] and for all rational functions at all values of [latex]a[/latex] for which the rational function is defined.

Limits of Polynomial and Rational Functions

Let [latex]p(x)[/latex] and [latex]q(x)[/latex] be polynomial functions. Let [latex]a[/latex] be a real number. Then,

[latex]\underset{x\to a}{\lim}p(x)=p(a)[/latex]

 

[latex]\underset{x\to a}{\lim}\dfrac{p(x)}{q(x)}=\dfrac{p(a)}{q(a)} \, \text{when} \, q(a)\ne 0[/latex]

To see that this theorem holds, consider the polynomial [latex]p(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots +c_1x+c_0[/latex]. By applying the sum, constant multiple, and power laws, we end up with

[latex]\begin{array}{cc}\hfill \underset{x\to a}{\lim}p(x)& =\underset{x\to a}{\lim}(c_nx^n+c_{n-1}x^{n-1}+\cdots +c_1x+c_0)\hfill \\ & =c_n(\underset{x\to a}{\lim}x)^n+c_{n-1}(\underset{x\to a}{\lim}x)^{n-1}+\cdots +c_1(\underset{x\to a}{\lim}x)+\underset{x\to a}{\lim}c_0\hfill \\ & =c_na^n+c_{n-1}a^{n-1}+\cdots +c_1a+c_0\hfill \\ & =p(a)\hfill \end{array}[/latex]

 

It now follows from the quotient law that if [latex]p(x)[/latex] and [latex]q(x)[/latex] are polynomials for which [latex]q(a)\ne 0[/latex], then

[latex]\underset{x\to a}{\lim}\dfrac{p(x)}{q(x)}=\dfrac{p(a)}{q(a)}[/latex]

The example below applies this result.

Example: Evaluating a Limit of a Rational Function

Evaluate the [latex]\underset{x\to 3}{\lim}\dfrac{2x^2-3x+1}{5x+4}[/latex].

Try It

Evaluate [latex]\underset{x\to -2}{\lim}(3x^3-2x+7)[/latex].

Watch the following video to see the worked solutions to all examples and try it’s on this page.