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

Infinite Limits The limit statement such as means that the function f increases without bound as x approaches c from either side, while means that the function g decreases without bound as x approaches c from either side.

Example 1 Sketch a graph of a function with the following characteristics: The graph has discontinuities at x = -2, 0, and 3. Only x = 0 is removable.

Example 2 Use the graph and complete the table to find the limit (if it exists). x 1.9 1.99 1.999 2 2.001 2.01 2.1 f(x) -100 -10000 -1000000 DNE -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.

Example 3 Use the graph and complete the table to find the limit (if it exists). x 1.9 1.99 1.999 2 2.001 2.01 2.1 f(x) -10 -100 -1000 DNE 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.

One-Sided Infinite Limits do Exist Example 4 Use the graph and complete the table to find the limits (if they exist). x 1.9 1.99 1.999 2 2.001 2.01 2.1 f(x) -10 -100 -1000 DNE 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.

The Existence of a Vertical Asymptote If is continuous c around and g(x) ≠ 0 around c, then x = c is a vertical asymptote of h(x) if f(c) ≠ 0 and g(c) = 0. 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

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

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

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

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