Find the Asymptotes y=x/(x^2+16)

Problem

y=x/(x2+16)

Solution

1. Identify vertical asymptotes by setting the denominator equal to zero.

x2+16=0

x2=−16

Since there are no real solutions to this equation, the function has no vertical asymptotes.

2. Identify horizontal asymptotes by finding the limit of the function as x approaches infinity or negative infinity.

(lim_x→∞)(x/(x2+16))

3. Compare the degrees of the numerator and the denominator. The degree of the numerator is 1 and the degree of the denominator is 2

1<2

4. Apply the rule for horizontal asymptotes where the degree of the denominator is greater than the degree of the numerator.

(lim_x→∞)(x/(x2+16))=0

(lim_x→−∞)(x/(x2+16))=0

5. Determine slant asymptotes by checking if the degree of the numerator is exactly one higher than the degree of the denominator. Since 1<2 there are no slant asymptotes.

Final Answer

Horizontal Asymptote: *y=0, Vertical Asymptotes: None

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