Chapter 2 – Limits and Continuity

2.1 Rates of Change and Limits Example: A rock breaks loose from the top of a tall cliff. What is its average speed during the first two seconds of fall?

Example: Find the speed of the rock in the last example at the instant t = 2. Def of a Limit

Properties of Limits

Polynomial and Rational Functions

Example: Calculate the following limits.

Ex: Use a graph to show that the following function does not exist.

One-Sided and Two-Sided Limits

Let’s examine the following graph to further explore right and left hand limits.

Find the following limits from the given graph. 4 o 1 2 3

Sandwich Theorem

Ex: Show that

Homework: p.66 (1- 63) odd

Explore problem number 75 on page 68

2.2 Limits Involving Infinity

[-6,6] by [-5,5] Ex: Find

Same properties hold for infinite limits as before. Ex: Find

Example: Let and . Show that while f and g are quite different for numerically small values of x, they are virtually identical for x large.

Let . Show that g(x) = x is a right end behavior model for f while is a left end behavior model for f.

Example “Seeing” Limits as x→±∞

Homework: p. 76 (1 – 55) odd

2.3 Continuity o

Continuity at a Point If a function f is not continuous at a point c , we say that f is discontinuous at c and c is a point of discontinuity of f. Note that c need not be in the domain of f.

Homework: p. 84 (1-43) odd

2.4 Rates of Change and Tangent Lines Example: Finding Average Rate of Change (Review) Find the average rate of change of f(x) = x3 - x over the interval [1,3]. Page 88

Let f(x) = 1/x Find the slope of the curve at x = a. Where does the slope equal -1/4? What happens to the tangent to the curve at the point (a,1/a) for different values of a?

Def: The normal line to a curve at a point is the line perpendicular to the tangent at that point. Ex: Write an equation for the normal to the curve f(x) = 4 – x2 at x = 1.

Homework: p. 92 (1 – 31)odd