To analyze the function \( f(x) = \log_4(x) \) for its asymptotic behavior, we need to consider its properties regarding horizontal and vertical asymptotes:
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Vertical Asymptote: The function \( \log_4(x) \) is defined only for \( x > 0 \). As \( x \) approaches 0 from the right (i.e., \( x \to 0^+ \)), \( f(x) \) approaches \(-\infty\). Therefore, there is a vertical asymptote at \( x = 0 \).
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Horizontal Asymptote: As \( x \) goes to infinity (i.e., \( x \to \infty \)), \( \log_4(x) \) goes to infinity as well. Thus, there is no horizontal asymptote, as the function does not approach a finite value.
Based on this analysis, the correct conclusion is:
B. The function f(x) has a vertical asymptote at x=0 and no horizontal asymptote.