Question

the graph of the function has a vertical asymptote of x = \boxed{\quad}
the graph of the function has a horizontal asymptote of y = \boxed{\quad}
done
$f(x)=\frac{1}{x}$

Explanation:

Step1: Find vertical asymptote

For the function \(y = \frac{1}{x}\), the denominator cannot be zero. When \(x = 0\), the function is undefined. As \(x\to0^{+}\), \(y\to+\infty\); as \(x\to0^{-}\), \(y\to-\infty\). So the vertical asymptote is \(x = 0\).

Step2: Find horizontal asymptote

We use the limit. Calculate \(\lim_{x\to\pm\infty}\frac{1}{x}\).
For \(\lim_{x\to+\infty}\frac{1}{x}\), as \(x\) gets larger and larger, \(\frac{1}{x}\) approaches \(0\).
For \(\lim_{x\to-\infty}\frac{1}{x}\), as \(x\) gets more and more negative (with large absolute value), \(\frac{1}{x}\) approaches \(0\). So the horizontal asymptote is \(y = 0\).

Answer:

The graph of the function has a vertical asymptote of \(x = 0\).
The graph of the function has a horizontal asymptote of \(y = 0\).