1.2 An Introduction to Limits

We have a point discontinuity at x = 1. What happens as from the left and from the right? x f(x) ? as x approaches 1 f(x) aproaches 3

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

Ex. Evaluate the function at several points near x = 0 and use the results to find the limit. x f(x) ? f(x) approached 2

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!!!

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.

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

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:

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 + )

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

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

Finding a for a given.

For all x in the interval (1,3)