1.3: Evaluating Limits Analytically Limitations live only in our minds. But if we use our imaginations, our possibilities become limitless.

Properties of Limits See Theorem 1.1 and 1.2

1.Limits of Polynomials 1) Try direct substitution first, if real number then done!

2) If direct substitution fails: Simplify to find another function that agrees for all x, except x = c Find:

Evaluate the following limits.

Summary – to find limits analytically 1)Try direct substitution 2)Simplify to a single fraction 3)Try factoring/cancel terms 4)Multiply by conjugate 1)Try direct substitution 2)Simplify to a single fraction 3)Try factoring/cancel terms 4)Multiply by conjugate Note: If none of these work, a graph or numerical investigation may give you insight into whether the limit exists at all (recall the 3 cases!)