LIMITS INVOLVING INFINITYSection 2.2a

Limits Involving Infinity

We can say “the limit of f as x approaches infinity,”meaning the limit of f as x moves increasingly far to theright on the number line, or…

Note: “Infinity” does not represent a real number……however,

Saying “the limit of f as x approaches negative infinity”means the limit of f as x moves increasingly far to theleft.

Limits Involving InfinityThe graph of the reciprocal function:

1f xx

Our new limits:

1lim 0x x

1lim 0

x x

The line y = 0 is a horizontalasymptote of the graph of f…

Definition: Horizontal AsymptoteThe line y = b is a horizontal asymptote of the graphof a function y = f(x) if either

or limx

f x b

limx

f x b

Ex: Find any H.A. of the graph of 2 1f x x

lim lim 2x x

f x f x

H.A.: y = 2

Definition: Horizontal AsymptoteThe line y = b is a horizontal asymptote of the graphof a function y = f(x) if either

or limx

f x b

limx

f x b

Ex: Find any H.A. of the graph of 2 1

xf xx

lim 1x

f x

H.A.: y = –1, y = 1

Investigate with both agraph and a table… lim 1

xf x

Sandwich Theorem Revisited

Find limx

f x

for sin xf xx

First, what do the graph and table suggest???

1 sin 1x Confirm Analytically:

1 sin 1xx x x

For x > 0, we have

1 sin 10 lim lim lim 0x x x

xx x x

And by the Sandwich Theorem:

Limits Involving InfinityProperties of Our New Limits:Note: All of the properties for limits approaching realnumbers also hold for limits approaching infinity!!!

Including Sum Rule, Difference Rule, Product Rule,Constant Multiple Rule, Quotient Rule, Power Rule

Limits Involving Infinity

Find5 sinlim

x

x xx

Rewrite:

5 sinlimx

x xx x

sinlim5 limx x

xx

5 0 5

Limits Involving InfinitySometimes, a function outgrows all bounds (eitherpositive or negative) as x approaches a finite numbera we write:

limx a

f x

or limx a

f x

Think back to the reciprocal function:

0

1limx x

and0

1limx x

The line x = 0 is a vertical asymptote of the graph of f…

Definition: Vertical AsymptoteThe line x = a is a vertical asymptote of the graphof a function y = f(x) if either

or limx a

f x

limx a

f x

Limits Involving InfinityFor the given function, (a) find the vertical asymptotes; (b) describethe behavior of the function to the left and right of each V.A.

2

12 5 3

xf xx x

Now, check the graph!V.A.:1 , 32

x x

0.5limx

f x

0.5limx

f x

1

2 1 3x

x x

3

limx

f x

3

limx

f x