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Understanding Infinite Limits and Vertical Asymptotes in Calculus

Infinite limits occur when the value of a fraction approaches a very large number as its denominator approaches zero. This section covers vertical asymptotes, which are defined as lines where the function tends toward infinity. We explore the properties of infinite limits, including behavior with positive or negative denominators. Additionally, we present examples such as the vertical asymptote at x = 0 for y = 1/x² and discuss how to identify vertical asymptotes and points of discontinuity. Understanding these concepts is crucial for mastering calculus.

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Understanding Infinite Limits and Vertical Asymptotes in Calculus

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  1. Section 1.5 - Infinite Limits

  2. 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.

  3. 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:

  4. 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.

  5. 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

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

  7. Properties of Infinite Limits 1. Sum or difference 2. Product 3. Quotient 0

  8. 0 HW Pg. 88 1-4, 29-51 odds, 61

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