1. 1.3 Evaluating Limits Analytically

2. Direct Substitution
If the the value of c is contained in the domain (the function exists at c) then
Direct Substitution is valid for ALL polynomials and rational functions with non-zero denominators

3. Find

4. Properties Of Limits
Basic - let b and c be real numbers and n be a positive integer
Constant
Identity
Power

5. Properties Of Limits
Let L, K, b, and, c be real numbers, let n be a positive integer, and
Scalar Multiple:
Sum/Difference:

6. Properties Of Limits
Let L, K, b, and, c be real numbers, let n be a positive integer, and
Power:
Product:

7. Properties Of Limits
Let L, K, b, and, c be real numbers, let n be a positive integer, and
Quotient:

8. Technique 1: Rewrite the function by factoring out Common factors
Find
Technique 1: Rewrite the function by
factoring out Common factors

9. Technique 2: Rationalize the numerator
Find
Technique 2: Rationalize the numerator
By multiplying by the complex conjugate

10. Technique 3: Use algebra to rewrite the the function
Find
Technique 3: Use algebra to rewrite the
the function

11. Strategies for Limits
Determine by recognition whether a limit can be evaluated by direct substitution
If direct substitution fails, try to use some technique (cancellation, rationalization, or algebraic manipulation)
Use a graph or table to verify your conclusion

12. Find

13. Find

14. 9) Use and

15. Homework
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# 5 – 25 odd, 37, 38, 39, odd, and assignment 1-3