Direct substitution

What you’ll learn

• Using direct substitution to find limits
• Applying limit laws to simplify expressions

Evaluating limits analytically

While graphs and tables are excellent tools for visualizing behavior or double-checking your work, most limits can be evaluated analytically using algebra and calculus. For many functions, straightforward techniques such as direct substitution and the basic limit laws can be used to determine a limit.

Direct substitution

The simplest way to evaluate a limit is direct substitution. If plugging in $x=a$ into $f(x)$ gives a finite number, then that number is the limit.

Always try direct substitution first.

Example 1. Evaluate

$$\underset{x\to -3}{\lim }(x^2+x)$$

Example 2. Evaluate

$$\underset{x\to 5}{\lim }\frac{x+1}{\sqrt{x-1}}$$

Limit properties

Limits follow a set of rules, called limit laws, that allow complicated limits to be rewritten as simpler ones.

Let $\underset{x\to a}{\lim }f(x)=L$ and $\underset{x\to a}{\lim }g(x)=M$. Then,

1. Constant multiple: If $c$ is a constant,

$$\underset{x\to a}{\lim }[c\times f(x)]=c\times L$$

2. Sum or difference:

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

3. Product:

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

4. Quotient:

$$\underset{x\to a}{\lim }\frac{f(x)}{g(x)}=\frac{L}{M},M\ne 0$$

5. Power:

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

Example 1. Given:

$$\begin{gathered} \underset{x\to 5}{\lim }f(x)=2 \\ \underset{x\to 5}{\lim }g(x)=4 \end{gathered}$$

Evaluate:

$$\underset{x\to 5}{\lim }[f(x)-3g(x){]}^2$$

Example 2. If

$$\begin{array}{rl}\underset{x\to -2}{\lim }f(x)& =-3\\ \underset{x\to -2}{\lim }g(x)& =6\end{array}$$

evaluate

$$\underset{x\to -2}{\lim }\frac{\sqrt{g(x)-f(x)}}{f(x{)}^2}$$

Using graphs

Limit laws can also be used after determining limit values from a graph.

Using the following graphs of $f(x)$ and $g(x)$, find

$$\underset{x\to 2}{\lim }\frac{\sqrt{f(x)+1}}{g(x)}$$