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1.3 Estimating Limit Values from Graphs

Syllabus
2020
Topic
1.3
Level
—

Learning objectives

—LIM-1.C—Estimate limits of functions• LIM-1.C Estimate limits of functions.• LIM-1.C.1 The concept of a limit includes one sided limits.• LIM-1.C.2 Graphical information about a function can be used to estimate limits.• LIM-1.C.3 Because of issues of scale, graphical representations of functions may miss important function behavior.• LIM-1.C.4 A limit might not exist for some functions at particular values of x. Some ways that the limit might not exist are if the function is unbounded, if the function is oscillating near this value, or if the limit from the left does not equal the limit from the right.• Enduring understanding LIM-1: Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

Estimating Limits from a Graph

To estimate lim⁡x→cf(x)\lim_{x\to c}f(x) from a graph, follow the graph's yy-values as xx moves toward cc. The plotted value f(c)f(c) does not determine the limit; the nearby behavior does.

\lim_{x\to c^-}f(x)=L_- \qquad \lim_{x\to c^+}f(x)=L_+

  1. Approach cc along the graph from the left and estimate L−L_-.
  2. Approach cc from the right and estimate L+L_+.
  3. If both sides approach the same finite value LL, conclude lim⁡x→cf(x)=L\lim_{x\to c}f(x)=L. If they do not agree or do not settle, the two-sided limit does not exist as a finite real number.
Graph behavior near x=cx=c Two-sided conclusion
Left and right both approach the same finite LL lim⁡x→cf(x)=L\lim_{x\to c}f(x)=L
Left and right approach different values Limit does not exist
Values become unbounded No finite real limit
Values keep oscillating without settling Limit does not exist

Suppose both branches of a graph approach height 33 as xx approaches 22, while a filled point is plotted at (2,5)(2,5). Then lim⁡x→2f(x)=3\lim_{x\to2}f(x)=3, even though f(2)=5f(2)=5.

A graph gives an estimate, not unlimited precision. Its window or scale can hide a small jump, rapid oscillation, or other local behavior, so do not claim more accuracy than the graph supports.