Limits

a limit is the value that a function or sequence "approaches" as the input approaches some value.

Fill in the blank…. You can always make a ___________ turn on red unless a sign is posted prohibiting it.

Give me liberty or ___________ me death!

Don’t cry over spilled ____________.

xf(x) Undefined 33 43

Let the graph below represent the function f(x)

Two important ideas about limits: The limit of a function at a point is the logical y-value at that point. We do not care what the value of a function actually is at the point where we’re looking for the limit.

Find the

Find

Try on your own by sketching a graph Find 1) 2) 3)

Three types of discontinuity Removable discontinuity – When the graph is continuous except for a hole. Jump discontinuity – When the left and right side limits are different. Infinite discontinuity – Created by vertical asymptotes.

Limits and Infinity Cheat code: Divide each term by x of the highest degree in the function.

The graph f(x) has, at most, one horizontal asymptote. – If the degree of the numerator (p(x)) is less than the degree of the denominator (q(x)), then the line y = 0 (the x-axis) is a horizontal asymptote. – If the degree of p(x) is equal to the degree of q(x), then the line y = a/b, where a is the leading coefficient of p(x) an b is the leading coefficient of q(x). – If the degree of p(x) > degree of q(x), then there are no horizontal asymptotes.

Limits Practice with solutions /limits.15/