Calculus AB: Applications of the Derivative

Problem : Find any vertical asymptotes for f (x) = .

Solution for Problem 1 >>

A candidate for a vertical asymptote is the place where the denominator goes to zero, which in this case is x = 3 . We must take limits to prove that this is an asymptote.

		= - ∞
		= + ∞

This means that x = - 3 is a vertical asymptote of f .

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Problem : Find any vertical asymptotes for f (x) = .

Solution for Problem 2 >>

		=
		= (x + 2) = 5
		=
		= (x + 2) = 5

This point is not an asymptote, but merely a point of discontinuity in the graph.

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Problem : Find any horizontal asymptotes for f (x) = .

Solution for Problem 3 >>

To find horizontal asymptotes, we take limits at infinity:

		=
		= = 1
		=
		= = 1

So, the line y = 1 is a horizontal asymptote of this function at both positive and negative infinity.

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Problem : Find any horizontal asymptotes for f (x) = .

Solution for Problem 4 >>

		=
		= = + ∞
		=
		= = - ∞

So, f does not have any horizontal asymptotes.

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