Euclid was a Greek mathematician best known for his treatise on geometry: The Elements . This influenced the development of Western mathematics for more than 2000 years.

Euclid was a Greek mathematician best known for his treatise on geometry: The Elements . This influenced the development of Western mathematics for more than 2000 years.

Euclid of Alexandria325 – 265 B.C.

Euclid of Alexandria325 – 265 B.C.

So we say,

This limit shows that f (x) hasa horizontal asymptoteat y = 1.

We take limits to infinity to analyze end-behavior of functions.

Left-Hand BehaviorRight-Hand Behavior

1f x

x

As the denominator gets larger, the value of the fraction gets closer to zero. So by definition, y = 0 is a horizontal asymptote.

1lim 0x x

1

lim 0x x

0

1limx x

0

1limx x

Recall

vertical asymptote at x = 0.

horizontal asymptote at y = 0.

Method I Method II

2

2

2

2

3 2lim

5 4 1

3lim

53

lim5

3

5

x

x

x

x x

x x

x

x

Algebraic Limits

5 4 2

2

2 1Find lim

3 5 7x

x x x

x x

Dominating term effect (DTE) we get

5

2

2lim

3x

x

x

32lim

3x

x

dominant terms in numerator and denominator

1Find lim tanx

x

1

So, and are2 2

horizontal asymptotes for

tan .

y y

y x

1tany x

Recall:

2 x x

if 0x x

if 0x x

Example

x

13 13

23 ( 23) 23

2lim

1x

x

x

2limx

x

x

This number becomes insignificant as .I call this the dominating term effect.

x

limx

x

x1

There is a horizontal asymptote at y = 1.

2Find lim

1x

x

x

lim , since 0x

xx

x

2( Note: )x x

sinlimx

x

x Find:

1 sin 1x

so for :0x 1 sin 1x

x x x

1 sin 1lim lim limx x x

x

x x x

sin0 lim 0

x

x

x

Thus, by the Squeeze Theorem:

sinlim 0x

x

x

x

y

When we graph this function, the limit appears to be zero.

5 sinlimx

x x

x

Find:

5 sinlimx

x x

x x

sinlim 5 limx x

x

x

5 0

5

5 sinlimx

x x

x

1lim sinx x

0

sin(0)

0

1Find lim sin

x x

21 5Find lim

3 7x

x

x

Find lim tanx

x

2Find lim 3 1x

x x x

4

2 2

5 7Find lim

2 1 3 2x

x

x x

2

2Analyze this ( ) !

1

xf x

x

a. Find the domain of f (x).

b. Determine any asymptotes for f (x).

c. Where is f (x) not continuous?