Section 1.5 - Infinite Limits

Section 1.5 - Infinite Limits

Infinite Limits:

1f xx

0

1limx x

As the denominator approaches zero, the value of the fraction gets very large.

If the denominator is positive then the fraction is positive.

0

1limx x

If the denominator is negative then the fraction is negative.

vertical asymptote at x=0.

Infinite limitsDefinition: The notation

(read as “the limit of f(x) , as x approaches a, is infinity”)means that the values of f(x) can be made arbitrarily large by

taking x sufficiently close to a (on either side of a) but not equal to a.

Note: Similar definitions can be given for negative infinity and the one-sided infinite limits.

)(lim xfax

Example:

20

1lim xx

Example:

20

1limx x

20

1limx x

The denominator is positive in both cases, so the limit is the same.

20

1 limx x

The key to thinking about this isthat as the denominator in a fractiongets larger, the fraction gets smallerand as the denominator gets smaller,the fraction gets larger.

So as the denominator gets infinitesimally small (towards 0), the fraction gets infinitesimally large (∞) .

Vertical AsymptotesDefinition: The line x=a is called a vertical asymptote of the

curve y=f(x) if at least one of the following statements is true:

)()()(

)()()(

limlimlimlimlimlim

xfxfxf

xfxfxf

axaxax

axaxax

Example: x=0 is a vertical asymptote for y=1/x2

Determine all vertical asymptotes and pointdiscontinuities of the graph of

Note: we have avertical asymptoteat x = 1 and a pointdiscontinuity at x = -3

lim as x 1 from L&R?

lim as x 1?

Properties of Infinite Limits

1. Sum or difference

2. Product

3. Quotient 0

0

HW Pg. 88 1-4, 29-51 odds, 61