1. Limits and Their Properties

2. Limits We would like to the find the slope of the tangent line to a curve… We can’t because you need TWO points to find a slope…

3. Instead, we use the slope of the SECANT line because two points are available. As the slope of the SECANT line approaches the slope of the TANGENT line, we are finding the LIMIT!

4. 3 cases where a limit DNE… *You may not have TWO values as a limit *Increasing without bounds *Constantly moving between TWO points.

5. Limits-> Evaluated by Substitution 1. Polynomials 2. Radicals 3. Rational Expressions…..ALL CONTINUOUS EVERYWHERE WHEN GRAPHED

6. If Direct Substitution Fails… 1. Factor, then cancel. 2. Rationalize the numerator. Ex:Ex:

7. Two Special Trig Limits… -Direct Substitution yields Undefined denominator. -Correct the limit as needed.

8. Continuity A graph is continuous if… 1. No gaps 2. No holes 3. No jumps

9. One Sided Limits Evaluate from the LEFT and the RIGHT Both limits MUST BE EQUAL in order for the limits to exist! **Both limits =

10. Infinite Limits A limit in which f(x) increases or decreases without bound as “x” approaches “c”.

11. To Find and Asymptote 1. Set the denominator equal to “zero” and solve 2. Answers are where vertical asymptotes exist. Ex: Vertical Asymptotes @ x = 4 and x = 1.

12. To Find Infinite Limits 1. Factor numerator and/or denominator if possible. 2. Cancel, if possible. 3. With what remains: A. Set numerator equal to zero to find x- intercepts. B.. Set denominator equal to zero to find vertical asymptotes 4. Select appropriate points to find designated limits.