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