Graph from assigned limits

Text

Sketch the graph of a function $f(x)$ whose domain is all of $\mathbb{R}$ (except possibly isolated excluded points) satisfying every condition below: $\lim_{x\to3^-}f=3^+,\qquad \lim_{x\to3^+}f=-\infty,$ $\lim_{x\to-3^-}f=-2^+,\qquad \lim_{x\to-3^+}f=4^-,$ $\lim_{x\to-\infty}f=+\infty,\qquad \lim_{x\to+\infty}f=-4^-.$

Solution

We translate each condition into a graph feature.

• $\lim_{x\to3^+}f=-\infty$: at $x=3$ there is a vertical asymptote (approaching from the right the graph drops to $-\infty$); from the left instead $\lim_{x\to3^-}f=3^+$, so the left branch approaches the ordinate $3$ (from above).
• At $x=-3$ the two one-sided limits are finite but different: $-2^+$ from the left and $4^-$ from the right. There is a jump (first-kind discontinuity) between $y=-2$ and $y=4$.
• $\lim_{x\to+\infty}f=-4^-$: horizontal asymptote $y=-4$ as $x\to+\infty$, approached from below.
• $\lim_{x\to-\infty}f=+\infty$: on the left the branch rises to $+\infty$.

A consistent graph (qualitative sketch):

$\boxed{\text{vertical as. } x=3;\ \text{jump at } x=-3\ (-2\to4);\ \text{horizontal as. } y=-4;\ \text{branch to }+\infty\text{ as }x\to-\infty}$