Exercise · ZW-ASYMPTOTES-003 · ★★☆ Intermediate
What are the asymptotes of \(f(x)=\frac{x^{2} + 1}{x^{2} - 4}\)?
Choose one option.
Show answer
- D. Vertical: \(x=-2\), \(x=2\); horizontal: \(y=1\)
Explanation
Vertical asymptotes are where the denominator vanishes (without the numerator vanishing): x = -2, x = 2. The horizontal one comes from \(\lim_{x\to\infty}f(x)=1\), that is, y = 1.
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-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-008 Which statement about f(x)=(x)/(x^2+1) is correct?
- ZW-ASYMPTOTES-009 What does lim _x→∞f(x)=L mean?