AP Calculus Review First Semester Differentiation to the Edges of Integration Sections , 3.9, (7.7)

Limit Definition (1.2) The number L is the limit of the function f(x) as x approaches c if, as the values of x get arbitrarily close (but not equal) to c, the values of f(x) approach (or equal) L:

When Limits Fail to Exist (1.2) Unbounded Behavior Oscillating Behavior

Evaluating Limits Analytically (1.3) Review the properties of limits, p. 57, if you dont know:

Strategies for Evaluating Limits (1.3 ) Ask yourself if the function can be evaluated by direct substitution (see previous slide) If it cannot, you can try –Cancelling and then using direct substitution –Using LHopitals Rule, provided you have 0/0

Strategies for Evaluating Limits, contd. (1.3 ) Or you can try – rationalizing and then using direct substitution –Using LHopitals Rule, provided you have 0/0

Evaluating Limits Analytically (1.3), contd. –You can also try using one of the two special trig limits: –For example: –Note: You could also use lHopitals Rule here.

Questions on Limits Which limits exist? Only this one

Questions on Limits Find the limit

Questions on Limits

Continuity (1.4) A function is continuous over an interval if we can draw its graph without lifting pencil from paper, i.e., the graph has no holes, breaks or jumps over the interval. More formally, the function y = f(x) is continuous at x = c if

Continuity, contd. (1.4) If then all three of the following are true:

Destroying Continuity (1)

Continuity, contd. (1.4) If then all three of the following are true:

Destroying Continuity (2)

Continuity, contd. (1.4) If then all three of the following are true:

Destroying Continuity (3)

Continuity, contd. (1.4) If then all three of the following are true:

Types of Discontinuity Removeable: mere point discontinuity Non-removeable: infinite discontinuity (vertical asymptotes)

Questions on Continuity 0

The function f is continuous at the point (c, f(c)). Which of the following statements could be false? This one

Intermediate Value Theorem If f is continuous on [a, b] and k is any number between f(a) and f(b), then there is a least one number c in [a, b] such that f(c) = k

Infinite Limits (1.5) A limit in which f(x) increases or decreases without bound as x approaches c is called an infinite limit. If f(x) approaches +/- infinity as x approaches c from the right or left, then the line x=c is a vertical asymptote of the graph of f.