Limits and Their Properties

LimitsWe would like to the find the slope of the

tangent line to a curve…

We can’t because you need TWO points to find a slope…

Instead, we use the slope of the SECANT line because two points are available.

As the slope of the SECANT line approaches the slope of the TANGENT line, we are finding the LIMIT!

3 cases where a limit DNE…lim0

xDNE

xx

20

1limx

DNEx

0

1limsinx

DNEx

*You may not have TWO values as a limit

*Increasing without bounds

*Constantly moving between TWO points.

Limits-> Evaluated by Substitution1. Polynomials2. Radicals3. Rational Expressions…..ALL CONTINUOUS

EVERYWHERE WHEN GRAPHED

2

2

2

lim4 3

4(2) 3

4(4) 3

16 3

19

xx

If Direct Substitution Fails…1. Factor, then cancel.2. Rationalize the numerator.

Ex: Ex: 2

5

5

5

25lim

5( 5)( 5)

lim5

lim( 5)

5 5

10

x

x

x

xxx xx

x

0

0

0

0

1 1 1 1lim

1 11 1

lim( 1 1)

lim( 1 1)

1lim( 1 1)

1 12( 0 1 1)

x

x

x

x

x xx xx

x x

x

x x

x

Two Special Trig Limits…sinlim 10

xxx

1 coslim 00

xxx

0

sin 2lim0sin 2

2lim2

2(1)2

x

xxxxx

-Direct Substitution yields Undefined denominator.-Correct the limit as needed.

ContinuityA graph is continuous if…1. No gaps2. No holes3. No jumps

One Sided LimitsEvaluate from the LEFT and the RIGHTBoth limits MUST BE EQUAL in order for

the limits to exist!

25

5lim

25x

x

x

25

5lim

25x

x

x

**Both limits = 110

Infinite LimitsA limit in which f(x) increases or decreases

without bound as “x” approaches “c”.

To Find and Asymptote1. Set the denominator equal to “zero” and

solve2. Answers are where vertical asymptotes

exist.Ex:

4( )2 5 4

2 5 4 0( 4)( 1) 04 0, 1 04, 1

f xx x

x xx xx xx x

Vertical Asymptotes @ x = 4 and x = 1.

To Find Infinite Limits1. Factor numerator and/or denominator if

possible.2. Cancel, if possible.3. With what remains:

A. Set numerator equal to zero to find x- intercepts.

B.. Set denominator equal to zero to find vertical asymptotes

4. Select appropriate points to find designated limits.