1. Evaluating Limits Analytically
Chapter 2.5

2. Basic Properties and Rules
Constant rule
Limit of x rule
Scalar multiple rule
Sum rule (the limit of a sum is the sum of the limits)

3. What stipulation must be made concerning D(x)?
Limits of Functions
Limit of a polynomial P(x)
Can be demonstrated using the basic properties and rules
Similarly, note the limit of a rational function
What stipulation must be made concerning D(x)?

4. Limit of a horizontal asymptote

5. General Strategies Verify the limit exist (graphically)
Substitute in the value to find the limit.
If you get an indeterminate form, try to cancel out a term in the numerator and denominator.
Multiply by the conjugate of a term.

6. Try It Out
Evaluate the limits by substitution 4

7. Some Examples Consider Strategy: simplify the algebraic fraction
Why is this difficult?
Strategy: simplify the algebraic fraction

8. Reinforce Your Conclusion
Graph the Function
Trace value close to specified point
Use a table to evaluate close to the point in question

9. Squeeze Rule (more to say about this later)
Given g(x) ≤ f(x) ≤ h(x) on an open interval containing c And …
Then

10. Some Examples