1.2 Finding Limits

1. 1.2 Finding Limits

2. Limits described • Goal: To see how a function behaves near a point (such as a discontinuity) use 2 sets of x-values that approach from the left and the right of that point • Definition: if f(x) approaches a #, L, as x approaches c from either side, the limit of f(x) as x approaches c is L.

3. Numerical estimation • Estimate the limit numerically by completing the table for

4. Answer • Limit is L=1 because y-values on both sides of x=2 approach y=1

5. Graphical interpretation • Determine the limits at x=1, 2, 3, and 4 given the graph • Remember limits occur at and when the y-values are the same as both sides of the c (x-values) approach c; otherwise L=DNE (does not exist)

6. Answer • The limits for the various c values: • C=1: L=1 (both sides approach y=1 despite point at y=2) • C=2: L=DNE (left is at y=2 and right is at y=3) • C=3: L=1 (both sides approach y=1) • C=4: L=2 (both sides approach y=2)

7. Show DNE • Show limit DNE for • Make a table or graph.

8. Answer • Left L=-1 and right L=1 • -1≠1 so L=DNE • Typical of step functions

9. Show DNE • Show limit DNE for • Make a table or graph.

10. Answer • Both sides approach positive infinity which is not a #; therefore, L=DNE • Typical of rational functions

11. Show DNE • Show limit DNE for • Make a table or graph.

12. Answer • F(x) oscillates between -1 and 1 so L=DNE • Typical of odd functions

13. Common behavior associated with DNE limits • F(x) approaches different # from right of c than approaches from left of c. • F(x) increases or decreases without bound as x approaches c. • F(x) oscillates between 2 fixed values as x approaches c.