Sketching a function from its limits (II)

Text

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

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

Solution

Reading the conditions:

• at $x=-1$ the left branch goes to $-\infty$ (vertical asymptote from the left), while the right branch tends to the value $2$ from below ($2^-$): so there is a jump at $x=-1$;
• as $x\to-\infty$ the function grows without bound ($+\infty$);
• as $x\to+\infty$ it approaches $y=3$ from below, so $y=3$ is a horizontal asymptote on the right.

One possible graph: a left branch descending from $+\infty$ (on the far left) down to $-\infty$ near $x=-1$; a right branch starting just below $y=2$ right after $x=-1$ and rising toward $y=3$ while staying below it. The solution is not unique.