Sketching a function from its asymptotes

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Sketch a function $h(x)$ with the following asymptotes:

• (a) vertical asymptote at $x=2$;
• (b) horizontal asymptote $y=-1$ (as $x\to-\infty$);
• (c) oblique asymptote $y=x+3$ (as $x\to+\infty$).

Solution

The graph must have three distinct asymptotic behaviours:

• near $x=2$ the curve diverges (to $+\infty$ or $-\infty$): vertical asymptote $x=2$;
• as $x\to-\infty$ it flattens onto the horizontal line $y=-1$;
• as $x\to+\infty$ it runs alongside the line $y=x+3$ (slope $1$, intercept $3$).

A function may stay near $y=-1$ on the left, diverge around $x=2$, and on the right grow following the line $y=x+3$. The solution is not unique (the horizontal asymptote on the left and the oblique one on the right can coexist since they apply to different branches).