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 cc

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

limx c

f (x)

limx c

f (x)

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)

f

f (x)

limx c

f (x) limx c

f (x) L

Consider

( | )a c b

( | )a c b

( | )a c b

f (c) is undefined

)(lim)(lim xfxfcxcx

limx cf (x) f (c)

limx c

f (x) limx c

f (x)

1) Find

limx 0

x x

x

limx 0

x x1

2

x1

2

limx 0

x1

2 x1

2 1

x1

2

1

21

21

21

)0(

1)0()0(

limx1f (x), if f (x)

x 3 1, x < 1

x +1, x 1

Left Right

2) Find

2) Find

limx1f (x), if f (x)

x 3 1, x < 1

x +1, x 1

limx1

x 3 1

Left Right

2

limx1

x 1

2

limx1f (x) 2

By existence theorem

f (x) x 2

x 2

2 x,

2

)2(

2 x,2

)2(

x

xx

x

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

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

f (x) x 2

x 2

limx 2

(x 2)

x 2

Left Right

1

limx 2

(x 2)

x 2

1

limx 2

f (x) DNE

(x 2)

x 2, x 2

(x 2)

x 2, x > 2

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

f (c)

limx cf (x)

limx cf (x) f (c)

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

f (x) x, x < 1

3, x 1

2x 1, x 1

f (1)a. exists

b.

limx1

f (x) limx1

(2x 1)

1

limx1

f (x) limx1

x

1

3

c.

limx1f (x) f (c)

Removable Point Discontinuity

5) Find the x - values at which is not continuous,

is the discontinuity removable?

f (x) x

x 2 1

)1(1)(

xx

xxf

(x 1)(x 1) 0

x 1,x 1

Non-removable: asymptotes

6) Find the x - values at which

is not continuous,

is the discontinuity removable?

f (x) x 3

x 2 9

f (x) x 3

(x 3)(x 3)

f (x) 1

(x 3)

Non-removable

x 3

Removable

x 3

7) If is continuous at ,

then

f (x) x 2 9

x 3

x 3

f ( 3)

f (x) (x 3)(x 3)

x 3

f ( 3) 3 3

f ( 3) 6

HOMEWORK

• Page 79 # 1-11, 18, 19, and 20