Sketching a graph from limit conditions (make-up exercise)

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Sketch the graph of a function $f(x)$ satisfying: $\lim_{x\to-\infty}f(x)=+\infty,\qquad \lim_{x\to0^-}f(x)=-\infty,\qquad \lim_{x\to0^+}f(x)=+\infty,$ $\lim_{x\to3^-}f(x)=2,\qquad \lim_{x\to3^+}f(x)=-1,\qquad \lim_{x\to+\infty}f(x)=-2^-.$

Solution

• $\lim_{x\to-\infty}f(x)=+\infty$: on the left the graph diverges to $+\infty$.
• $\lim_{x\to0^-}f(x)=-\infty$ and $\lim_{x\to0^+}f(x)=+\infty$: a vertical asymptote $x=0$, with the left branch plunging to $-\infty$ and the right branch rising to $+\infty$.
• $\lim_{x\to3^-}f(x)=2$ and $\lim_{x\to3^+}f(x)=-1$: at $x=3$ the two one-sided limits are finite but different — a jump (discontinuity of the first kind) from the value $2$ (from the left) to the value $-1$ (from the right).
• $\lim_{x\to+\infty}f(x)=-2^-$: a right horizontal asymptote $y=-2$, approached from below.

$\boxed{\text{asymptotes } x=0,\ y=-2;\ \text{jump at }x=3}$