Find the horizontal and vertical asymptote(s) of the following function.

Question

Find the horizontal and vertical asymptote(s) of the following function.

y = 2e^x / (e^x − 5)

Answer 1

The vertical asymptote occurs when the denominator equals 0.

e^x - 5 = 0 means e^x = 5, or x = ln(5)

For the horizontal asymptotes, we get one as x -> ∞ and one as x -> -∞

As x -> ∞, multiplying by e^-x /e^-x we get lim x -> ∞ 2e^x / (e^x − 5) = lim x -> ∞ 2/(1 - 5e^-x ) = 2/(1-0) = 2

As x -> -∞, we get lim x -> -∞ 2e^x / (e^x − 5) = 0/(0-5) = 0

Answer 2

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