Limits and Their Properties

Limits - Part 1

Multiple Choice

Warm-up:

For  $$f\left(x\right)\ =\ x^2\ -\ 4$$ , what is  $$f\left(2\right)$$  ? 

• 0

• 4

• -4

• 8

Definition of a Limit

Let f(x) be a function and let a and L be real numbers such that f(x) is defined for all x around a, but not necessarily including a.

If, as x approaches the value of a, we have that f(x) approaches the value of L, then the limit of the function as x approaches a is L. We write this as

 $\lim_{x\rightarrow a}f\left(x\right)\ =\ L$  .

*Note: There are a couple of methods for finding limits that are found in the textbook, such as by calculator approximation, but we will not be covering them.

What Does That Mean?

In essence, we want to know what a function "looks like it is doing", as opposed to knowing what the function actually does at that point.

 $y\ =\ x^2\ -\ 4$  

As x gets closer to the value of 2, we can see on the graph that the function gets closer to 0.

When it comes to graphs and limits, ask yourself, "What does the function look like it's doing?".

Multiple Choice

Evaluate $$\lim_{x\rightarrow4}\ f\left(x\right)$$  .

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• 1

• 0

• The limit does not exist.

Multiple Choice

Evaluate $$\lim_{x\rightarrow3}\ f\left(x\right)$$  .

• 4

• 1

• 0

• The limit does not exist.

Multiple Choice

Evaluate $$\lim_{x\rightarrow1}\ g\left(x\right)$$  .

• 6

• The limit does not exist because the function is undefined at x = 1.

• 0

• -6

Multiple Choice

Evaluate $$\lim_{x\rightarrow5}\ g\left(x\right)$$  .

• 6

• The limit does not exist.

• 0

• -6

Limit Rules

The next slide has various limit rules for us to look over. We are not going to go into depth on these rules, but we should recognize them.

Limit Rules

*Note: As Rule 7 suggests, your first step in evaluating a limit is to plug it in and see what happens. As long as you don’t get $\frac{0}{0}$  

  or  $\frac{\infty}{\infty}$  , you will have an answer you can work with.

   $\frac{0}{0}$  and   $\frac{\infty}{\infty}$  are called indeterminant forms.

Open Ended

This concludes part one of our lesson on limits. We will into the algebra of limits tomorrow. Before we wrap up, what concerns or questions do you have over limits so far? Type your response below.