Exercise · ZW-ASYMPTOTES-008 · ★★★ Advanced
Which statement about \(f(x)=\frac{x}{x^{2}+1}\) is correct?
Choose one option.
Show answer
- C. It has horizontal asymptote \(y=0\) and the graph crosses it at \(x=0\)
Explanation
The denominator never vanishes, so there are no vertical asymptotes. Since \(\lim_{x\to\pm\infty}f(x)=0\), the horizontal one is y = 0, and f(0)=0 crosses it at the origin.
Review the concept: Asymptotes
An asymptote is a line the graph approaches without touching it. The line \(x=a\) is vertical if \(f\) tends to \(\pm\infty\) as \(x\to a\); the line \(y=L\) is horizontal if \(f(x)\to L\) as \(x\to\pm\infty\).
\[\begin{gathered}x=a:\ \lim_{x\to a}f(x)=\pm\infty\\ y=L:\ \lim_{x\to\pm\infty}f(x)=L\end{gathered}\]
More practice problems: asymptotes
- ZW-ASYMPTOTES-001 What are the asymptotes of f(x)=(2 x + 1)/(x - 3)?
- ZW-ASYMPTOTES-002 What are the asymptotes of f(x)=(3 x)/(x + 2)?
- ZW-ASYMPTOTES-003 What are the asymptotes of f(x)=(x^2 + 1)/(x^2 - 4)?
- ZW-ASYMPTOTES-004 What are the asymptotes of f(x)=(1)/(x+5)?
- ZW-ASYMPTOTES-005 What is a vertical asymptote of f at x=a?
- ZW-ASYMPTOTES-006 What are the asymptotes of f(x)=(x-1)/(x^2-1)?
- ZW-ASYMPTOTES-007 What are the asymptotes of f(x)=(2)/((x-1)^2)?
- ZW-ASYMPTOTES-009 What does lim _x→∞f(x)=L mean?