6. Intuitive Limits and Continuity

We begin our study of Calculus with Limits. In this chapter we want to develop an intuitive understanding of limits. In a later chapter, we will get a more precise definition.

a. What my Algebra Teacher Never Told me!

We begin with a multiple choice question you might have gotten in a high school algebra class:

Simplify the function $f(x)=\dfrac{x^2-4}{x-2}$ .

$f(x)=x-2$ $f(x)=x+2$ Undefined None of these

Answer

Incorrect. Sorry that's not correct. Factor the numerator and cancel!

[×]

Answer

Incorrect. This is the answer you would have given in a high school algebra class, but it is not correct in a calculus class. If we factor the numerator and cancel we appear to get $\dfrac{x^2-4}{x-2}=\dfrac{(x+2)(x-2)}{x-2}=x+2$ However, we cannot just cancel.
Notice $f(x)=x+2$ is defined for all $x$ , but, $f(x)=\dfrac{x^2-4}{x-2}$ is undefined at $x=2$ because we would be dividing by $0$ . If we cancel, we lose the fact that $f(x)$ is undefined at $x=2$ .

[×]

Answer

Incorrect. You are on the right track, but $f(x)$ is defined for most $x$ .

[×]

Answer

Correct! $f(x)=x+2$ is defined for all all $x$ . However, $f(x)=\dfrac{x^2-4}{x-2}$ is undefined at $x=2$ because we would be dividing by $0$ . So a correct simplification would be: $\begin{aligned} f(x)&=\dfrac{x^2-4}{x-2}=\dfrac{(x+2)(x-2)}{x-2} \\ &=\left\{\begin{matrix} x+2 & \text{if} & x\ne2 \\ \text{undefined} & \text{if} & x=2 \end{matrix}\right. \end{aligned}$

[×]

$\leftarrow\leftarrow\leftarrow$ Be sure to read this after you do the exercise!

So a correct simplification in the above exercise is: $\begin{aligned} f(x)&=\dfrac{x^2-4}{x-2}=\dfrac{(x+2)(x-2)}{x-2} \\ &=\left\{\begin{matrix} x+2 & \text{if} & x\ne2 \\ \text{undefined} & \text{if} & x=2 \end{matrix}\right. \end{aligned}$ A plot of $f(x)$ is shown at the right.

eg_never_told — Graph of $f(x)=\dfrac{x^2-4}{x-2}$ .

The graph of $y=x+2$ is a straight line with slope $m=1$ and $y$ -intercept $b=2$ as shown at the right.
The graph of $y=\dfrac{x^2-4}{x-2}$ is the same straight line but with a hole at $x=2$ as shown above.

eg_never_told_wrong — Graph of $y=x+2$ .

If we look at the graph, we see that as $x$ approaches $x=2$ , the value of $y$ approaches $y=4$ . (This is the value we get if we plug $x=2$ into $y=x+2$ .) To capture this idea of approaching $y=4$ even though we can never actually get there, we define the limit: $\lim_{x\to2}f(x) =\lim_{x\to2}\dfrac{x^2-4}{x-2} =4$ This (intuitive) definition will be made clearer on the next page.

[×]

6. Intuitive Limits and Continuity

a. What my Algebra Teacher Never Told me!

Answer

Answer

Answer

Answer

Heading