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1.15 Connecting Limits at Infinity and Horizontal Asymptotes

Syllabus
2020
Topic
1.15
Level
—

Learning objectives

—LIM-2.D—Interpret the behavior of functions using limits involving infinity—Topic 1.15• LIM-2.D Interpret the behavior of functions using limits involving infinity.• LIM-2.D.3 The concept of a limit can be extended to include limits at infinity.• LIM-2.D.4 Limits at infinity describe end behavior.• LIM-2.D.5 Relative magnitudes of functions and their rates of change can be compared using limits.• Enduring understanding LIM-2: Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

End Behavior, Horizontal Asymptotes, and Relative Growth

A limit at infinity describes one end of a graph: x→∞x\to\infty follows the graph far to the right, while x→−∞x\to-\infty follows it far to the left. If the outputs approach a finite number LL on either end, y=Ly=L is a horizontal asymptote for that end.

\lim_{x\to\infty}f(x)=L\quad\text{or}\quad\lim_{x\to-\infty}f(x)=L;\Longrightarrow;y=L\text{ is a horizontal asymptote}

For f(x)=3x2−xx2+4f(x)=\dfrac{3x^2-x}{x^2+4}, divide numerator and denominator by x2x^2: f(x)=3−1/x1+4/x2→3f(x)=\dfrac{3-1/x}{1+4/x^2}\to3 as x→±∞x\to\pm\infty. Thus both ends approach the horizontal asymptote y=3y=3.

lim⁡x→∞f(x)g(x)\lim_{x\to\infty}\dfrac{f(x)}{g(x)} Relative magnitude for large positive xx
00 gg dominates ff
Finite nonzero CC The functions have comparable magnitude; ff is approximately CgCg
±∞\pm\infty ff dominates gg in magnitude

For f(x)=x2f(x)=x^2 and g(x)=x3g(x)=x^3, f(x)/g(x)=1/x→0f(x)/g(x)=1/x\to0, so x3x^3 grows faster in magnitude. A horizontal asymptote controls end behavior only: the graph may cross it at finite xx, and the two ends may have different limiting values.