Unit 1: Limits and Continuity

1.1 Intro to limits

• A limit is one of the most fundamental concepts of calculus. The limit of a function $f$ at $x=a$ is the numerical value $f$ approaches as we get closer and closer to $x=a$ .

• We can write limits in limit notation as $\lim_{x\to a}f\left(x\right)$ , which is pronounced “the limit of $f\left(x\right)$ as $x$ approaches $a$ .

• Graphed to the right is the function $y=t\left(x\right)$ . As an example, lets approximate $\lim_{x\to2}t\left(x\right)$ .

• Recall that $\lim_{x\to2}t\left(x\right)$ is the numerical value that $t$ approaches as $x$ gets closer and closer to $2$ . Examining the graph, we can see that $\lim_{x\to2}t\left(x\right)=3.1$ .

• However, it might also be tempting to say that $\lim_{x\to2}t\left(x\right)=1$ because $t\left(2\right)=1$ . This is a common fallacy many students make when evaluating limits—we must always remember that $\lim_{x\to a}f\left(x\right)\ne f\left(a\right)$ .

• When we evaluate limits, we want to think about what happens as $x$ gets infinitely close. What this means is that we can always get as close to the limit as we want.

• For example, if we want to find a limit $\lim_{x\to7}f\left(x\right)$ , we want to think about $f\left(6.9\right),f\left(6.99\right),f\left(6.999\right)\ldots$ and so on. We also want to think about $f\left(7.1\right),f\left(7.01\right),f\left(7.001\right)\ldots$ and so on.

• Lastly, for a limit to exist, it must approach the same value from both sides. If a limit $\lim_{x\to a}f\left(x\right)$ approaches a different value from the left of $a$ than from the right of $a$ , then we can say that $\lim_{x\to a}f\left(x\right)$ does not exist.

1.2 Estimating limits from graphs

• Earlier, we introduced the idea that $\lim_{x\to a}f\left(x\right)\ne f\left(a\right)$ . Let’s reinforce this idea by examining two graphs. In both examples, we will attempt to find the limit $\lim_{x\to1}$ .

• An unbounded limit occurs when we attempt to take the limit of a vertical asymptote. At a vertical asymptote, the function we try to take the limit of skyrockets to infinity without ever touching the $x$ -value it is at. So, when we try to approximate what a function approaches at a vertical asymptote, we will find that the limit does not exist because the function never approaches anything.

• We can see an unbounded limit in the example above when we try to take the limit $\lim_{x\to0}h\left(x\right)$ . The function never approaches anything, so $\lim_{x\to0}h\left(x\right)$ does not exist.

• Earlier, we introduced the idea that if a limit approaches two different values from both sides, it is undefined. We can rewrite this idea by saying that a limit is undefined if its two one-sided limits are different. Lets discuss what a one-sided limit is:

• A positive one-sided limit describes a limit $\lim_{x\to a}f\left(x\right)$ from values that are greater than $a$ , or from the right of $a$ . We denote positive one-sided limits with a superscript “+” sign in the limit notation, as $\lim_{x\to a^{+}}f\left(x\right)$ .

• On the flip side, a negative one-sided limit describes a limit $\lim_{x\to a}f\left(x\right)$ from values that are less than $a$ , or from the left of $a$ . We denote negative one-sided limits with a superscript “-” sign in the limit notation, as $\lim_{x\to a^{-}}f\left(x\right)$ .

• Looking at our example $g\left(x\right)$ (which is shown to the left for convenience) from above, lets try to find the one-sided limits at $x=1$ .

• When we approach $x=1$ from the left, $g\left(x\right)$ approaches $0$ . So, $\lim_{x\to1^{-}}g\left(x\right)=0$ .

• When we approach $x=1$ from the right, $g\left(x\right)$ approaches $-1$ . So, $\lim_{x\to1^{+}}g\left(x\right)=-1$ .

• What we said above can now be written as “$\lim_{x\to a}f\left(x\right)$ is undefined if $\lim_{x\to a^{+}}f\left(x\right)\ne\lim_{x\to a^{-}}f\left(x\right)$ .”

  
  

1.3 Estimating limits from tables

• Before we begin estimating limits from tables, we need to be able to accurately create appropriate tables for approximating limits. There are a few things that we need to make sure we consider, listed below:

  • We need to simulate getting infinitely close to the limit by selecting values that approach the limit from both sides. As an example/reminder, if we want to evaluate a limit $\lim_{x\to0}f\left(x\right)$ , we want to consider $f\left(0.1\right),f\left(0.01\right),f\left(0.001\right)\ldots$ and so on. We also want to consider $f\left(-0.1\right),f\left(-0.01\right),f\left(-0.001\right)\ldots$ and so on.

    • This also means not approaching the limit in constant increments. We want to decrease the distance to the point of interest, not keep it the same.

  • We also want to make sure that we approach the limit from both sides. This means that we need to consider