Limits read from a graph (asymptotes x=−2, x=3, y=4)

Text

From the graph below write the relevant limits (one-sided limits at the vertical asymptotes and limits at infinity). Finally give the value of the horizontal asymptote $y_{as}$.

Solution

We read the behaviour near each vertical asymptote and at infinity.

• At $x=-2$: from the left the branch rises to $+\infty$, from the right it drops to $-\infty$.
• At $x=3$: from the left it rises to $+\infty$, from the right it drops to $-\infty$.
• As $x\to+\infty$ the graph approaches the line $y=4$ (horizontal asymptote).
• As $x\to-\infty$ the branch rises to $+\infty$.

$\boxed{\begin{aligned}&\lim_{x\to-2^-}f=+\infty,\ \lim_{x\to-2^+}f=-\infty,\\ &\lim_{x\to3^-}f=+\infty,\ \lim_{x\to3^+}f=-\infty,\\ &\lim_{x\to+\infty}f=4,\ \lim_{x\to-\infty}f=+\infty\end{aligned}}$

The horizontal asymptote is $y_{as}=4$.