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Section 1.5: Infinite Limits

This guide explores infinite limits and vertical asymptotes in calculus. A vertical asymptote at x = c indicates that a function f(x) approaches infinity or negative infinity as x approaches c from either side. The behavior of f(x) around a vertical asymptote determines if an infinite limit exists. We provide examples to illustrate the identification of vertical asymptotes by analyzing numerator and denominator conditions, along with the use of graphs. Learn the big ideas that define the existence of vertical asymptotes and their properties.

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Section 1.5: Infinite Limits

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  1. Section 1.5: Infinite Limits

  2. Vertical Asymptote If f(x) approaches infinity (or negative infinity) as x approaches c from the right or the left, then the line x = cis a vertical asymptote of the graph of f.

  3. Infinite Limits The function f increases without bound as x approaches c from either side The function gdecreases without bound as x approaches c from either side.

  4. Example 1 Use the graph and complete the table to find the limit (if it exists). -100 -10000 DNE -1000000 -1000000 -10000 -100 If the function behaves the same around an asymptote, then the infinite limit exists. The function decreases without bound as x approaches 2 from either side.

  5. Example 2 Use the graph and complete the table to find the limit (if it exists). -10 -100 DNE -1000 1000 100 10 If the function behaves the different around an asymptote, then the infinite limit does not exist. The function increases without bound as x approaches 2 from the right and decreases without bound as x approaches 2 from the left.

  6. Example 3 Use the graph and complete the table to find the limits (if they exist). -10 -100 DNE -1000 1000 100 10 One-Sided Infinite Limits do Exist The function increases without bound as x approaches 2 from the right and decreases without bound as x approaches 2 from the left.

  7. The Existence of a Vertical Asymptote Big Idea: x = c is a vertical asymptote if c ONLY makes the denominator zero. Ex: Determine all vertical asymptotes of . When is the denominator zero: Do the x’s make the numerator 0? No for both x=1 and x=-1 are vertical asymptotes Must have equations for asymptotes If is continuous around c and g(x) ≠ 0 around c, then x = cis a vertical asymptote of h(x) if f(c) ≠ 0 and g(c) = 0.

  8. Example 2 Determine all vertical asymptotes of . When is the denominator zero: Do the x’s make the numerator 0? Yes… No! x=2 is a vertical asymptote EXTRA: What about x = -1? Therefore, x=1 is a removable discontinuity

  9. Example 3 Analytically determine all vertical asymptotes of We Know: When is the denominator zero: Do the x’s make the numerator 0? No, since the numerator is a constant. 0 and π are angles that make sine 0 Find all of the values since trig functions are cyclic

  10. Example 3 Cont. Analytically determine all vertical asymptotes of Check with the graph

  11. Properties of Infinite Limits • Sum/Difference: • Product: • Quotient: • Example: Since , then Let c and L be real numbers and f and g be functions such that:

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