1.1 A Preview of Calculus and 1.2 Finding Limits Graphically and Numerically

Objectives Understand what calculus is and how it compares to precalculus. Estimate a limit using a numerical or graphical approach. Learn different ways that a limit can fail to exist.

Swimming Speed Swimming Speed: Taking it to the Limit Questions 1-5

Preview of Calculus Diagrams on pages 43 and 44

Two Areas of Calculus: Differentiation Animation of Differentiation Animation of Differentiation

Two Areas of Calculus: Integration Animation of Integration Animation of Integration

Limits Both branches of calculus were originally explored using limits. Limits help define calculus.

1.2 Finding Limits Graphically and Numerically

Find the Limit x f(x) ? x approaches 1 from the left x approaches 1 from the right Limits are independent of single points.

Exploration (p. 48) From the graph, it looks like f(2) is defined. Look at the table. On the calculator: tblstart 1.8 and ∆Tbl=0.1. Look at the table again. What does f approach as x gets closer to 2 from both sides?

Example Look at the graph and the table.

Example Limits are NOT affected by single points!

Three Examples of Limits that Fail to Exist If the left-hand limit doesn't equal right-hand limit, the two-sided limit does not exist.

Three Examples of Limits that Fail to Exist If the graph approaches ∞ or -∞ from one or both sides, the limit does not exist.

Three Examples of Limits that Fail to Exist Look at the graph and table. As x gets close to 0, f(x) doesn't approach a number, but oscillates back and forth. If the graph has an oscillating behavior, the limit does not exist.

Limits that Fail to Exist f(x) approaches a different number from the right side of c than it approaches from the left side. f(x) increases or decreases without bound as x approaches c. f(x) oscillates as x approaches c.

Homework 1.2 (page 54) #5, 7, odd