1. Bell Ringer

2. Bell Ringer continued…
Does the limit exist at x = 1 if so what is it?

3. Finding Limits Analytically

4. Strategies for Finding Limits
To find limits analytically, try the following:
Direct Substitution (Always try this FIRST!!!)
If Direct Substitution fails, then rewrite then find a function that is equivalent to the original function except at one point. Then use Direct Substitution. Methods for this include…
Factoring/Dividing Out Technique
Rationalize Numerator/Denominator
Eliminating Embedded Denominators
Trigonometric Identities
Legal Creativity

5. The slides that follow investigate why Direct Substitution is valid.
One of the easiest and most useful ways to evaluate a limit analytically is direct substitution (substitution and evaluation):
If you can plug c into f(x) and generate a real number answer in the range of f(x), that generally implies that the limit exists (assuming f(x) is continuous at c).
Example:
Always check for substitution first.
The slides that follow investigate why Direct Substitution is valid.

6. Properties of Limits Constant Function Limit of x
Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:
Constant Function
Limit of x
Limit of a Power of x
Scalar Multiple

7. Properties of Limits Sum Difference Product Quotient Power
Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:
Sum Difference
Product
Quotient
Power

8. Example
Let and Find the following limits.

9. Example 2 Sum/Difference Property Multiple and Constant Properties
Direct Substitution
Power Property
Limit of x Property

10. Direct Substitution
Direct substitution is a valid analytical method to evaluate the following limits.
If p is a polynomial function and c is a real number, then:
If r is a rational function given by r(x) = p(x)/q(x), and c is a real number, then
If a radical function where n is a positive integer. The following limit is valid for all c if n is odd and only c>0 when n is even:

11. Direct Substitution
Direct substitution is a valid analytical method to evaluate the following limits.
If the f and g are functions such that Then the limit of the composition is:
If c is a real number in the domain of a trigonometric function then:

12. Example
Direct Substitution can be used since the function is well defined at x=3
For what value(s) of x can the limit not be evaluated using direct substitution?
At x=-6 since it makes the denominator 0:

13. Indeterminate Form
Evaluate the limit analytically:
An example of an indeterminate form because the limit can not be determined. 1/0 is another example.
Often limits can not be evaluated at a value using Direct Substitution. If this is the case, try to find another function that agrees with the original function except at the point in question. In other words…
How can we simplify: ?

14. REVIEW To find limits analytically, try the following:
Direct Substitution (Try this FIRST)
If Direct Substitution fails, then rewrite then find a function that is equivalent to the original function except at one point. Then use Direct Substitution. Methods for this include…
Factoring/Dividing Out Technique
Rationalize Numerator/Denominator
Eliminating Embedded Denominators
Trigonometric Identities
Legal Creativity

15. This function is equivalent to the original function except at x=2
Example 1
Evaluate the limit analytically:
Factor the numerator and denominator
At first Direct Substitution fails because x=2 results in dividing by zero
Cancel common factors
This function is equivalent to the original function except at x=2
Direct substitution

16. Cancel the denominators of the fractions in the numerator
Example 2
Evaluate the limit analytically:
Cancel the denominators of the fractions in the numerator
If the subtraction is backwards, Factoring a negative 1 to flip the signs
Cancel common factors
Direct substitution

17. Expand the the expression to see if anything cancels
Example 3
Evaluate the limit analytically:
Expand the the expression to see if anything cancels
Factor to see if anything cancels
Direct substitution

18. Rewrite the tangent function using cosine and sine
Example 4
Evaluate the limit analytically:
Eliminate the embedded fraction
Rewrite the tangent function using cosine and sine
If the subtraction is backwards, Factoring a negative 1 to flip the signs
Direct substitution

19. Example 5 Evaluate the limit analytically:
Rationalize the numerator by multiplying by conjugate
Cancel common factors
Direct substitution

25. The Sandwich Theorem