Question

which best describes the asymptote of an exponential function of the form ( f(x) = b^x )?

a. horizontal asymptote at ( y = 1 )

b. vertical asymptote at ( x = 1 )

c. vertical asymptote at ( x = 0 )

d. horizontal asymptote at ( y = 0 )

Explanation:

Step1: Recall Exponential Function Asymptote

For an exponential function \( f(x) = b^x \) (where \( b>0, b
eq1 \)), analyze the limit as \( x\to\pm\infty \).

• As \( x\to -\infty \), \( b^x=\frac{1}{b^{|x|}}\to 0 \) (since \( b^{|x|}\to\infty \) for \( b > 1 \), and \( b^{|x|}\to 0 \) for \( 0 < b < 1 \), but in both cases, \( \frac{1}{b^{|x|}} \) approaches 0).
• As \( x\to\infty \), \( b^x \) either grows without bound ( \( b > 1 \)) or approaches 0 ( \( 0 < b < 1 \)), but the horizontal asymptote is determined by the limit as \( x\to -\infty \) (or the behavior that the function approaches a constant as \( x \) moves in one direction).

Step2: Evaluate Options

• Option A: \( y = 1 \) is not the asymptote, since \( b^x \) approaches 0, not 1, as \( x\to -\infty \).
• Option B: Exponential functions have no vertical asymptotes (they are defined for all real \( x \)), so vertical asymptote at \( x = 1 \) is incorrect.
• Option C: No vertical asymptote at \( x = 0 \) (function is defined at \( x = 0 \), \( f(0)=1 \)), so this is incorrect.
• Option D: The limit as \( x\to -\infty \) of \( b^x \) is 0, so there is a horizontal asymptote at \( y = 0 \).

Answer:

D. Horizontal asymptote at \( y = 0 \)