Section Infinite Limits

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Presentation transcript:

Section 1.5 - Infinite Limits

Infinite Limits: As the denominator approaches zero, the value of the fraction gets very large. vertical asymptote at x=0. If the denominator is positive then the fraction is positive. If the denominator is negative then the fraction is negative.

Infinite limits Definition: 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. Example:

Example: The denominator is positive in both cases, so the limit is the same. So as the denominator gets infinitesimally small (towards 0), the fraction gets infinitesimally large (∞) . The key to thinking about this is that as the denominator in a fraction gets larger, the fraction gets smaller and as the denominator gets smaller, the fraction gets larger.

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

Determine all vertical asymptotes and point discontinuities of the graph of Note: we have a vertical asymptote at x = 1 and a point discontinuity at x = -3 lim as x 1? lim as x 1 from L&R?

Properties of Infinite Limits 1. Sum or difference 2. Product 3. Quotient

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