Find the Asymptotes f(x)=2^x

Problem

ƒ(x)=2^x

Solution

1. Identify the type of function. The function ƒ(x)=2^x is a basic exponential function with a base greater than 1

2. Check for vertical asymptotes. Exponential functions of the form b^x are defined for all real numbers x so there are no vertical asymptotes.

3. Analyze the end behavior as x approaches infinity. As x→∞ 2^x grows without bound toward ∞ so there is no horizontal asymptote on the right side.

4. Analyze the end behavior as x approaches negative infinity. As x→−∞ the expression becomes 2^(−∞) which is equivalent to 1/2^∞

5. Evaluate the limit. Since 1/2^∞ approaches 0 the function approaches the xaxis.

(lim_x→−∞)(2^x)=0

6. Determine the horizontal asymptote. Because the limit as x approaches negative infinity is 0 the line y=0 is a horizontal asymptote.

Final Answer

Horizontal Asymptote: *y=0

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