1. Sec. 1.3: Evaluating Limits Analytically
The limit of f(x) as x approaches c does not depend on the value of f at c.
i.e. The limit of f(x) as x approaches c may not be f(c).
Although, for those that are, we could have used direct substitution to evaluate the limit.

2. Limits Using Direct Substitution
If b and c are real numbers and n is a positive integer, then

3. More Limits Using Direct Substitution
If p is a polynomial function, then If r is a rational function r(x) = p(x)/q(x), then

4. More Limits Using Direct Substitution
For radical functions, if n is positive, then the following limit is valid for all c if n is odd, and all c > 0 if n is even.

5. More Limits Using Direct Substitution
For trigonometric functions, if c is in the domain of the function, then

6. Properties of Limits (Rules)
Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits.

7. More Properties of Limits
The limit of a composite function: If f and g are functions such that then

8. What if direct substitution won’t work?
Try the following:
If direct substitution gives 0/0, then
Factor and use the dividing out strategy.
Rationalize and use the dividing out strategy.
Simplify and use the dividing out strategy.
Still won’t work? Fall back on using a graph or a table.

9. Sandwich Theorem
If f(x)h(x) g(x) for all x in an open interval containing a, except possibly at a itself, and if then exists and is equal to L.

10. Example
Use the Sandwich Theorem to prove that