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1.14 Connecting Infinite Limits and Vertical Asymptotes

Syllabus
2020
Topic
1.14
Level
—

Reading Infinite Limits as Vertical Asymptotes

An infinite limit says that function values become unbounded as xx approaches a finite input. The symbol +∞+\infty means values grow without bound; −∞-\infty means they decrease without bound. Infinity is a direction of behavior, not a real number reached by the function.

Limit statement Graph behavior near x=ax=a
lim⁡x→a−f(x)=+∞\lim_{x\to a^-}f(x)=+\infty The left branch rises without bound as it approaches aa
lim⁡x→a−f(x)=−∞\lim_{x\to a^-}f(x)=-\infty The left branch falls without bound as it approaches aa
lim⁡x→a+f(x)=+∞\lim_{x\to a^+}f(x)=+\infty The right branch rises without bound as it approaches aa
lim⁡x→a+f(x)=−∞\lim_{x\to a^+}f(x)=-\infty The right branch falls without bound as it approaches aa

\lim_{x\to a^-}f(x)=\pm\infty\quad\text{or}\quad\lim_{x\to a^+}f(x)=\pm\infty;\Longrightarrow;x=a\text{ is a vertical asymptote}

For f(x)=1x−2f(x)=\dfrac{1}{x-2}, approaching 22 from the left makes x−2x-2 a small negative number, so f(x)→−∞f(x)\to-\infty. From the right, x−2x-2 is small and positive, so f(x)→+∞f(x)\to+\infty. Therefore x=2x=2 is a vertical asymptote even though the two sides head in opposite directions.

Do not write f(2)=∞f(2)=\infty. A vertical asymptote describes nearby unbounded behavior; the function may be undefined at the asymptote, and one infinite side is sufficient to identify it.