1. The Indeterminate Form
Some limits can be found by using direct substitution.
When using direct substitution, if the result is a finite number, then that number is the limit of the function.
If the result is in the form
, this is an Indeterminate Form.
In this case, one of the following techniques can be used:
Factor the expression, simplify, then use direct substitution.
Rationalize the numerator, simplify, then use direct substitution.
Simplify the complex fraction, then use direct substitution.

2. Factoring Example
Direct substitution yields an indeterminate form, so we must factor.
Find the limit of:
Factor
Simplify
Substitute

3. Rationalizing Example
Direct substitution yields an indeterminate form, so we must rationalize.
Find the limit of:
Rationalize
Simplify
Substitute

4. Complex Fraction Example
Direct substitution yields an indeterminate form, so we must simplify the complex fraction.
Find the limit of:
Simplify
Substitute

5. Homework
Page 67: 28 – 36
Page 68:
Even Numbers Only

6. Special Trigonometric Limits
Let’s see what the graph looks like.
Now let’s look at this graph.
As x approaches 0, f(x) approaches 1
As x approaches 0, f(x) approaches 0
That was easy

7. Different Formats of the Special Trigonometric Limits
Let’s look at those special limits one more time.
The symbol x can be replaced with any other expression as long as that expression also approaches 0.
As x approaches 0, 3x also approaches 0.
Factor out
Multiply both sides of the equation by 3.

8. Trigonometric Limit Examples
That’s just the upside-down version of the first one.
Hey, that looks familiar. I can do this.
We know that
If we factor, we can use this to solve our problem.
Multiply both sides by 4.
That was easy

9. The Squeeze Theorem If
for all x in an open interval containing c, except possibly at c itself, and if
then
Don’t worry, that just means if f(x) is between h(x) and g(x), and the limit of h(x) is equal to the limit of g(x), then the limit of f(x) is also that same limit. g f h c

10. Squeeze Theorem Example
Use a graphing calculator to graph each function and observe that the squeeze theorem exists.
Then find the limit as x approaches 0 analytically for each function.

11. Another Squeeze Theorem Example
Use the Squeeze Theorem to find

12. Homework
Page 68: 65 – 70
88 & 89