Find the Asymptotes f(x)=1/(x^2+1)

Problem

ƒ(x)=1/(x2+1)

Solution

1. Identify vertical asymptotes by finding values of x that make the denominator zero.

2. Solve the equation x2+1=0

3. Determine that x2=−1 has no real solutions, meaning there are no vertical asymptotes.

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

5. Evaluate the limit (lim_x→∞)(1/(x2+1))

6. Observe that as x becomes very large, the denominator x2+1 grows without bound, causing the fraction to approach 0

7. Conclude that the horizontal asymptote is y=0

Final Answer

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

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