To determine the asymptotes of the function [tex]\( g(x) = \frac{10}{x} \)[/tex], follow these steps:
### Vertical Asymptote
1. Identify where the function is undefined.
The function [tex]\( g(x) = \frac{10}{x} \)[/tex] is undefined when the denominator is zero.
Therefore, set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x = 0
\][/tex]
2. Determine the vertical asymptote.
The value of [tex]\( x \)[/tex] where the function is undefined is the location of the vertical asymptote.
Hence, the vertical asymptote is at [tex]\( x = 0 \)[/tex].
### Horizontal Asymptote
1. Analyze the behavior as [tex]\( x \)[/tex] approaches infinity or negative infinity.
The horizontal asymptote is determined by the value that [tex]\( g(x) = \frac{10}{x} \)[/tex] approaches as [tex]\( x \)[/tex] goes to infinity or negative infinity.
As [tex]\( x \)[/tex] approaches infinity ([tex]\( x \to \infty \)[/tex]), [tex]\( \frac{10}{x} \)[/tex] approaches 0.
Similarly, as [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]), [tex]\( \frac{10}{x} \)[/tex] also approaches 0.
2. Determine the horizontal asymptote.
Since [tex]\( g(x) \)[/tex] approaches 0 in both directions, the horizontal asymptote is at [tex]\( y = 0 \)[/tex].
### Final Answer
- The vertical asymptote is at [tex]\( x = 0 \)[/tex].
- The horizontal asymptote is at [tex]\( y = 0 \)[/tex].
