Calculus Cookbook: Chapter 2

1. Calculus Cookbook: Chapter 2 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 EQUALin 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.