Graph from assigned limits (2)

Text

Sketch the graph of a function $f(x)$ satisfying every condition: $\lim_{x\to-1^-}f=+\infty,\qquad \lim_{x\to-1^+}f=-1^-,$ $\lim_{x\to4^-}f=-\infty,\qquad \lim_{x\to4^+}f=3^-,$ $\lim_{x\to-\infty}f=2,\qquad \lim_{x\to+\infty}f=0^-.$

Solution

We interpret the conditions.

• At $x=-1$: from the left $+\infty$ (vertical asymptote for the left branch); from the right the limit is finite, $-1^-$ (the branch restarts just below $y=-1$). Discontinuity.
• At $x=4$: from the left $-\infty$ (vertical asymptote for the middle branch); from the right finite, $3^-$ (restarts just below $y=3$). Discontinuity.
• $\lim_{x\to-\infty}f=2$: horizontal asymptote $y=2$ on the left.
• $\lim_{x\to+\infty}f=0^-$: horizontal asymptote $y=0$ on the right, approached from below.

Qualitative sketch:

$\boxed{\text{vert. as. } x=-1,\ x=4;\ \text{horiz. as. } y=2\ (x\to-\infty),\ y=0^-\ (x\to+\infty);\ \text{jumps at } x=-1,\ x=4}$