To determine the horizontal asymptote of the function [tex]\( f(x) = \frac{5}{x} \)[/tex], we analyze the behavior of the function as [tex]\( x \)[/tex] approaches infinity or negative infinity.
1. Understanding Horizontal Asymptotes:
A horizontal asymptote of a function is a horizontal line [tex]\( y = c \)[/tex] where the function [tex]\( f(x) \)[/tex] approaches [tex]\( c \)[/tex] as [tex]\( x \)[/tex] tends to either positive infinity [tex]\( +\infty \)[/tex] or negative infinity [tex]\( -\infty \)[/tex].
2. Considering [tex]\( x \to \infty \)[/tex]:
We analyze what happens to [tex]\( f(x) = \frac{5}{x} \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex]:
[tex]\[
\lim_{{x \to +\infty}} \frac{5}{x} = 0
\][/tex]
As [tex]\( x \)[/tex] becomes very large, the value of [tex]\(\frac{5}{x}\)[/tex] becomes very small and approaches 0.
3. Considering [tex]\( x \to -\infty \)[/tex]:
Similarly, we consider the behavior as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex]:
[tex]\[
\lim_{{x \to -\infty}} \frac{5}{x} = 0
\][/tex]
As [tex]\( x \)[/tex] becomes very large in the negative direction, the value of [tex]\(\frac{5}{x}\)[/tex] also becomes very small and approaches 0.
4. Conclusion:
Since the value of [tex]\( f(x) \)[/tex] approaches 0 as [tex]\( x \)[/tex] approaches both [tex]\( +\infty \)[/tex] and [tex]\( -\infty \)[/tex], the horizontal asymptote of the function is [tex]\( y = 0 \)[/tex].
Therefore, the correct answer is [tex]\( \boxed{f(x) = 0} \)[/tex].
