1. 4. Algebraic Limits

2. So far ….. Numerically Graphically
What if you don’t have a graph or a calculator or your brilliant friend sitting next to you?
We will look at various algebraic methods. Their names are not important, but it is important to recognize their forms.

3. 1. Direct Substitution This is the method you should ALWAYS try first!
Example 1

4. 2. VA
If you use direct substitution and get , that means there is a VA and the answer will be DNE or if it is a one-sided limit
Example 2

5. Indeterminate form
We say that f(x) is indeterminate at x = c if when we evaluate f(c), we obtain an undefined expression
Strategy is to transform f(x) algebraically into a new expression that is defined and continuous at x=c.
Remember that if you get something like then this is NOT indeterminate

6. 3. Factor and divide out
The factor that causes 0/0 in the denominator can be simplified with its common factor in the numerator and then direct substitution can be used on the resulting function
Example 3

7. 4. Rationalization conjugation
This method works well where there is a sum or difference of one or more radical terms. It involves multiplying by conjugate/conjugate
Example 4

8. 5. Least common denominator
This method works well if there is a compound fraction or a complex fraction (a fraction within a fraction.) It involves finding the LCD
Example 5

9. 6. Expand
This method works well if there is some obvious math to do like expanding binomials
Example 6

10. 7. Trig Manipulation
We have already seen this method, it involves using trig identities to rewrite limits to be ones we have memorized
Example 7

11. 8. General cleverness
Try this when all else fails.
Example 8

12. or other indeterminant
summary
Real #
(this is your answer)
Compare
Exponents VA
Plug in # to left and/or right
Plug in
is answer
or other indeterminant
Algebra to simplify
Then plug in