Find the Asymptotes f(x)=xe^(-x)

Problem

ƒ(x)=x*e(−x)

Solution

1. Identify the domain of the function. Since x and e(−x) are defined for all real numbers, the domain is (−∞,∞) There are no vertical asymptotes because there are no values of x that make the function undefined or approach infinity.

2. Evaluate the limit as x approaches positive infinity to find horizontal asymptotes. Rewrite the expression to use L'Hôpital's Rule.

(lim_x→∞)(x)*e(−x)=(lim_x→∞)(x/(ex))

3. Apply L'Hôpital's Rule because the limit is in the indeterminate form ∞/∞

(lim_x→∞)(x/(ex))=(lim_x→∞)(d(x)/d(x)/d(ex)/d(x))

(lim_x→∞)(1/(ex))=0

This indicates a horizontal asymptote at y=0 as x→∞

4. Evaluate the limit as x approaches negative infinity.

(lim_x→−∞)(x)*e(−x)

As x→−∞ the term e(−x)→∞ Therefore:

(−∞)*(∞)=−∞

There is no horizontal asymptote as x→−∞

5. Check for slant (oblique) asymptotes by calculating the limit of ƒ(x)/x as x→±∞

(lim_x→∞)((x*e(−x))/x)=(lim_x→∞)(e(−x))=0

Since the limit is 0 which is the slope of our horizontal asymptote, there is no unique slant asymptote. As x→−∞ the limit (lim_x→−∞)(e(−x))=∞ so no slant asymptote exists in that direction either.

Final Answer

Horizontal Asymptote: *y=0

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