1.4 One-Sided Limits and Continuity. Definition A function is continuous at c if the following three conditions are met 2. Limit of f(x) exists 1. f(c)

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Presentation transcript:

1.4 One-Sided Limits and Continuity

Definition A function is continuous at c if the following three conditions are met 2. Limit of f(x) exists 1. f(c) is defined 3. Limit of f(x) is c c

Definition If a function is defined on an interval I, except at c, then the function is said to have a discontinuity at c such as a hole, break or asymptote

One-Sided Limits Approach a function from different directions both graphically and analytically 1)Limits from the right 2)Limits from the left

Existence of a Limit Let be a function and let c and L be real numbers. The limit of as x approaches c is L if and only if (iff)

Consider ( | ) a c b ( | ) a c b ( | ) a c b

1) Find

Left Right 2) Find

LeftRight By existence theorem

3) Determine if the limit exists at x = -2 if

Left Right

Continuity at a point Let f(x) be defined on an open interval containing c, f(x) is continuous at c if a. is defined (exists) b. exists (one-sided limits are equal) c.The

Discontinuity Removable: the function can be redefined (hole discontinuity)

Discontinuity Non - Removable: a.Jump - breaks at a particular value b. Infinite discontinuity - vertical asymptote

4)Find the x - values where is not continuous and classify a. exists b. c. Removable Point Discontinuity

5)Find the x - values at which is not continuous, is the discontinuity removable? Non-removable: asymptotes

6)Find the x - values at which is not continuous, is the discontinuity removable? Non-removable Removable

7)If is continuous at, then

HOMEWORK Page 79 # 1-11, 18, 19, and 20