Sketching a function from its limits

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Sketch the graph of a function $g(x)$ satisfying:

• (a) $\displaystyle\lim_{x \to 1} g(x) = 3$
• (b) $\displaystyle\lim_{x \to -2^-} g(x) = -\infty$
• (c) $\displaystyle\lim_{x \to -2^+} g(x) = +\infty$
• (d) $\displaystyle\lim_{x \to \infty} g(x) = 0$

Solution

The conditions translate into:

• a vertical asymptote at $x=-2$, with the curve going to $-\infty$ from the left and to $+\infty$ from the right;
• a limiting value $(1,3)$: approaching $x=1$ the curve tends to height $3$;
• a horizontal asymptote $y=0$ at the ends, so the curve flattens onto the $x$-axis as $x\to\pm\infty$.

One possible graph: a left branch dropping from the $x$-axis toward $-\infty$ as it approaches $x=-2$; a right branch descending from $+\infty$ (just after $x=-2$), passing through height $3$ near $x=1$ and then tending to $0$. The solution is not unique.