Finding Limits Graphically and Numerically

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Presentation transcript:

Finding Limits Graphically and Numerically 1.2 Finding Limits Graphically and Numerically An Introduction to Limits x y x y Sketch the graph of the function.

Finding Limits Graphically and Numerically 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.

Finding Limits Graphically and Numerically 1.2 Finding Limits Graphically and Numerically Example 1 Estimating a limit numerically Estimate the value of the following limit. x y 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.

Finding Limits Graphically and Numerically 1.2 Finding Limits Graphically and Numerically Example 2 Finding a limit Estimate the value of the following limit.

Finding Limits Graphically and Numerically 1.2 Finding Limits Graphically and Numerically Example 3 Behavior that differs from the right and left Estimate the value of the following limit.

Finding Limits Graphically and Numerically 1.2 Finding Limits Graphically and Numerically 10 11 12 9 8 6 7 -5 -4 4 5 6 3 2 -3 1 5 3 -6 4 -7 -8 -9 -4 -5 1 -3 2 -2 -1 Example 4 Unbounded behavior Estimate the value of the following limit.

Finding Limits Graphically and Numerically 1.2 Finding Limits Graphically and Numerically 12 -5 -4 11 9 7 8 -3 10 -2 4 5 6 3 2 -1 1 6 5 4 -4 -6 -7 -9 -8 -3 -5 2 -2 1 3 -1 Example 5 Oscillating behavior Estimate the value of the following limit. t f (t)

Finding Limits Graphically and Numerically 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

Finding Limits Graphically and Numerically 1.2 Finding Limits Graphically and Numerically 12 -5 -4 11 10 7 8 9 -3 -2 4 5 6 3 2 1 -1 6 5 -4 -6 -7 -9 -8 -3 -5 3 -2 2 4 1 -1 Example 6 Finding a d for a given e Given the limit find d such that whenever

Finding Limits Graphically and Numerically 1.2 Finding Limits Graphically and Numerically

1.1 A Preview of Calculus Pg. 46, 1.1 #1-11