Graph from assigned limits (3)

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Sketch the graph of a function $f(x)$ satisfying: $\lim_{x\to2^-}f=+\infty,\qquad \lim_{x\to2^+}f=3^+,$ $\lim_{x\to-\infty}f=+\infty,\qquad \lim_{x\to+\infty}f=-1^-.$

Solution

Graph features.

• At $x=2$: from the left $+\infty$ (vertical asymptote for the left branch); from the right the limit is finite, $3^+$ (the branch restarts just above $y=3$). Discontinuity.
• $\lim_{x\to-\infty}f=+\infty$: on the left the branch rises to $+\infty$; together with the limit at $2^-$ the left branch forms a “valley”.
• $\lim_{x\to+\infty}f=-1^-$: horizontal asymptote $y=-1$ on the right, approached from below.

Qualitative sketch:

$\boxed{\text{vert. as. } x=2\ (\text{left branch}\to+\infty);\ \text{horiz. as. } y=-1\ (x\to+\infty);\ \text{jump at } x=2\ (\text{from the right }3^+)}$