1.3 Limits, Continuity, & End Behavior September 21 st, 2015 SWBAT estimate a limit of a graphed function. SWBAT identity a point of discontinuity utilizing the continuity test.

What is a limit?

If the value of f(x) approaches a unique value L as x approaches c from each side, then the limit of f(x) as x approaches c equals L. The above symbol reads “ The limit of f(x) as x approaches c is L.”

Estimating the limit graphically

Estimating the Limit graphically

Continuity A graph of a continuous function has no breaks, holes, or gaps. You can trace the graph of a continuous function without lifting up your pencil. Continuity Test A function f (x) is continuous at x = c if it satisfies the following three conditions. 1. f (x) is defined at c. If the limit as x approaches c does not equal the value of the function at x = 3, the function is discontinuous at that point.

EXIT TICKET 1.Choose a value of x that is continuous for the function to the left and explain why the graph is continuous at that point. 2.Choose a value of x that is discontinuous for the function to the left and explain why the graph is discontinuous at that point.

1.3 Limits, Continuity, & End Behavior September 22nd, 2015 SWBAT identify the three types of discontinuity. SWBAT determine when the limit does not exist

THREE TYPES OF DISCONTINUITY: #1 REMOVABLE DISCONTINUITY A graph has a removable discontinuity if the function is continuous except for a hole at x = c.

THREE TYPES OF DISCONTINUITY #2 INFINITE DISCONTINUITY A function has an infinite discontinuity at x = c if the function value increases or decreases indefinitely as x approaches c from the left and right.

THREE TYPES OF DISCONTINUITY #3 JUMP DISCONTINUITY A function has a jump discontinuity at x = c if the limits of the functions as x approaches c from the left and the right exist but have two distinct values.

ONE SIDED LIMITS Left Hand Limit If the function approaches a unique value L as x approaches c from the left, then which is read, “The limit of f (x) as x approaches c from the left is L.” Right Hand Limit If the function approaches a unique value L as x approaches c from the right, then which is read, “The limit of f (x) as x approaches c from the right is L.”

When the limit does not exist

ENTRANCE TICKET

1.3 Limits, Continuity, & End Behavior September 23 rd, 2015 SWBAT determine a limit algebraically. SWBAT determine if a function is continuous at a point algebraically

Evaluating Limits & Continuity Algebraically METHOD #1 – DIRECT SUBSTITUTION Direct substitution can be used as long as the denominator does not equal zero when c is substituted. For instance, = 5(5) + 8 = = 33 Now prove that the functions in #1, #2, and #3 are continuous at point c. Yes, if you can use direct substitution then the limit of a function at x = c equals f(c), so the function is continuous at c.

Evaluating Limits & Continuity Algebraically Method #2 - Factoring Factor to simplify the function and eliminate the zero in the denominator 5. Determine if #1 through #4 are continuous at the given c.

Evaluating Limits & Continuity Algebraically Method #3 - Rationalizing Practice

Discontinuity Red Flag #1 Piece Wise Functions Just look at the jumps, but wait Look at x = 1 So Don’t Assume, Check

Discontinuity Red Flag #2 VARIABLE IN THE DENOMINATOR If the zero in the denominator can be eliminated by factoring, the function has a removable discontinuity. If the zero in the denominator cannot be eliminated by factoring, the function has an infinite discontinuity

1.3 Limits, Continuity, & End Behavior September 24 th, 2015 SWBAT graph a piece wise function given conditions concerning limits and continuity

Graph a function h(x) that satisfies the following conditions:

LIMITS APPROACHING INFINITY If you miss this day, you must copy notes from peers since the lecture was done on the white board.