1. 1.2 An Introduction to Limits

2. We have a point discontinuity at x = 1. What happens as from the left and from the right? x f(x).75.9.99.999 1 1.001 1.01 1.1 ?2.31 2.71 2.97 2.99 3.003 3.03 3.3 as x approaches 1 f(x) aproaches 3

3. Sometimes we can find a limit by just plugging in the number we are approaching. Ex. Find the limit.

4. Ex. Evaluate the function at several points near x = 0 and use the results to find the limit. x f(x) -.01 -.001 -.0001 0.0001.001.01 1.9949 1.9995 1.99995 ? 2.000052.00050 2.0049 f(x) approached 2

5. Ex. Find the limit as x 2 where f(x) = What is the y-value as x approaches 2 from the left and from the right? The limit is 1 since f(x) = 1 from the left and from the right as x approaches 2. The value of f(2) is immaterial!!!

6. 3 types of limits that fail to exist. 1.Behavior that differs from the left and from the right. Ex. 1 the limit D.N.E., since the limit from the left does not = the limit from the right.

7. 2.Unbounded behavior Ex. Since f(x) the limit D.N.E.

8. 3.Oscillating behavior (use calculator) As x 0, f(x) oscillates between –1 and 1, therefore the limit D.N.E. 1.f(x) approaches a different number from the right side of c than it approaches from the left side. 2. f(x) increases or decreases without bound as x approaches c. 3.f(x) oscillates between two fixed values as x approaches c. Limits D.N.E. when:

9. A Formal Definition of a Limit If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, then we say that the limit of f(x), as x approaches c, is L. (c,L) In the figure to the left, let represent a small positive number. Then the phrase “f(x) becomes arbitrarily close to L” means that f(x) lies in the interval (L -, L + )

10. Def. of a Limit Let f be a function defined on an open interval containing c and let L be a real number. The statement means that for each whenever

11. Finding a for a given. Given the limit: find To find delta, we establish a connection between Thus, we choose

12. Finding a for a given.

13. For all x in the interval (1,3)