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

Problem

y=(1/2)^x

Solution

1. Identify the type of function. The given equation y=(1/2)^x is an exponential function of the form y=a⋅b^x+k where a=1 b=1/2 and k=0

2. Determine the horizontal asymptote. For exponential functions, the horizontal asymptote is determined by the behavior of the function as x approaches infinity or negative infinity.

3. Evaluate the limit as x→∞ Since the base b=1/2 is between 0 and 1 the value of (1/2)^x approaches 0 as x becomes very large.

(lim_x→∞)(1/2)=0

4. Evaluate the limit as x→−∞ As x becomes a large negative number, the expression becomes 2^|x| which grows without bound.

(lim_x→−∞)(1/2)=∞

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

6. Conclude the horizontal asymptote. Based on the limit as x→∞ the graph approaches the line y=0

Final Answer

y=0

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