4. Algebraic Limits

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.

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

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

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

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

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

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

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

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

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

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