Infinite Limits Unit IB Day 5. Do Now For which values of x is f(x) = (x – 3)/(x 2 – 9) undefined? Are these removable or nonremovable discontinuities?

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Presentation transcript:

Infinite Limits Unit IB Day 5

Do Now For which values of x is f(x) = (x – 3)/(x 2 – 9) undefined? Are these removable or nonremovable discontinuities?

Ex. 1 Let f(x) = 3/(x – 2). ▫Find x f(x)f(x) 

Infinite Limits (Informal) A limit in which f(x) increases or decreases without bound as x approaches c is called an infinite limit. ▫Technically, the limit ______________!

Infinite Limits (Formal Definition) Let f be a function that is defined at every real number in some open interval containing c (except possibly at c itself). The statement means that for each M > 0 there exists a δ > 0 such that f(x) > M whenever 0 < |x – c| < δ.

Ex. 2: Infinite Limits from Graphs Use the graph to determine the limit of each function as x approaches 1 from the left and from the right. a) b)c) d)

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

Thm Vertical Asymptotes Let f and g be continuous on an open interval containing c. If f(c) ≠ 0, g(c) = 0, and there exists an open interval containing c such that g(x) ≠ 0 for all x ≠ c in the interval, then the graph of h(x) = f(x)/g(x) has a vertical asymptote at x = c.

Ex. 3: Finding Vertical Asymptotes Determine all vertical asymptotes of the graph of each function. a) b)

Ex. 4: Rational Function with Common Factors Determine all vertical asymptotes of the graph of

Ex. 5: Determining Infinite Limits Find each limit. a) b)