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

Problem

y=1/(x2+1)

Solution

1. Identify vertical asymptotes by finding values of x where the denominator is zero and the numerator is non-zero.

2. Solve the equation x2+1=0 for x

3. Determine real solutions for the denominator. Since x2+1=0 leads to x2=−1 there are no real solutions. Therefore, there are no vertical asymptotes.

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

5. Evaluate the limit (lim_x→∞)(1/(x2+1)) As x grows larger, the denominator grows larger, causing the fraction to approach 0

6. Evaluate the limit (lim_x→−∞)(1/(x2+1)) Similarly, as x approaches negative infinity, the fraction approaches 0

7. Conclude the horizontal asymptote is the line y=0

Final Answer

Horizontal: *y=0, Vertical: None

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