Question

1. which type of asymptote(s), if any, does the graph of $f(x) = \frac{1}{x}$ have?

only horizontal asymptotes
no asymptotes
both vertical and horizontal asymptotes
only vertical asymptotes

Explanation:

Step1: Find vertical asymptote

The function \( f(x) = \frac{1}{x} \) is undefined at \( x = 0 \) (denominator zero). As \( x \to 0^+ \), \( f(x) \to +\infty \); as \( x \to 0^- \), \( f(x) \to -\infty \). So vertical asymptote is \( x = 0 \).

Step2: Find horizontal asymptote

Compute the limit as \( x \to \pm\infty \): \( \lim_{x \to \pm\infty} \frac{1}{x} = 0 \). So horizontal asymptote is \( y = 0 \).

Step3: Conclusion

The function has both vertical and horizontal asymptotes.

Answer:

both vertical and horizontal asymptotes