1 2 Finding Limits Graphically and Numerically An

1. 2 Finding Limits Graphically and Numerically An Introduction to Limits Sketch the graph of the function. x y

1. 2 Finding Limits Graphically and Numerically An Introduction to Limits Definition of a limit: We say that the limit of f(x) is L as x approaches a and write this as provided we can make f(x) as close to L as we want for all x sufficiently close to a, from both sides, without actually letting x be a.

1. 2 Finding Limits Graphically and Numerically Example 1 Estimating a limit numerically Estimate the value of the following limit. x Limits are asking what the function is doing around x = a, and are not concerned with what the function is actually doing at x = a. y

1. 2 Finding Limits Graphically and Numerically Example 2 Finding a limit Estimate the value of the following limit.

1. 2 Finding Limits Graphically and Numerically Example 3 Behavior that differs from the right and left Estimate the value of the following limit.

1. 2 Finding Limits Graphically and Numerically 90 8 7 -1 -2 -6 -7 -8 -9 6 5 4 3 2 1 0 -3 -4 -5 12 11 Example 4 Unbounded behavior Estimate the value of the following limit.

1. 2 Finding Limits Graphically and Numerically 90 8 7 -6 -7 -8 -9 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 12 11 Example 5 Oscillating behavior Estimate the value of the following limit. t f (t)

1. 2 Finding Limits Graphically and Numerically The Formal Definition of a Limit Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then we say that, if for every number e > 0 there is some number d > 0 such that whenever

1. 2 Finding Limits Graphically and Numerically 90 8 7 -6 -7 -8 -9 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 12 11 Example 6 Finding a d for a given e find d such that Given the limit whenever

1. 2 Finding Limits Graphically and Numerically

1. 1 A Preview of Calculus Pg. 46, 1. 1 #1 -11