1.4 Estimating Limit Values from Tables

Learning objectives

—LIM-1.C—Estimate limits of functions—Topic 1.4• LIM-1.C Estimate limits of functions.• LIM-1.C.5 Numerical information can be used to estimate limits.• Enduring understanding LIM-1: Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

To estimate $\lim_{x\to c}f(x)$ numerically, inspect values of $f(x)$ for inputs increasingly close to $c$ from below and from above. A common output trend on both sides gives the two-sided estimate.

Choose several $x$ -values less than $c$ and several greater than $c$ .
Make the inputs successively closer to $c$ without relying only on $x=c$ .
Compare the output trend from the two sides.
If both approach the same value $L$ , estimate $\lim_{x\to c}f(x)\approx L$ .

For $f(x)=x^2$ near $x=2$ :

$x$	$1.9$	$1.99$	$1.999$	$2.001$	$2.01$	$2.1$
$f(x)$	$3.61$	$3.9601$	$3.996001$	$4.004001$	$4.0401$	$4.41$

From the left, the outputs rise toward $4$ ; from the right, they fall toward $4$ . Therefore the table supports $\lim_{x\to2}x^2=4$ . The closest inputs usually provide the strongest estimate because they show the local trend near $2$ .

A finite table supports an estimate but does not by itself prove a limit. Widely spaced or one-sided inputs can hide a jump, oscillation, or other behavior closer to $c$ ; use values from both sides and state only the precision supported by the data.

Next Topic

—1.5 Determining Limits Using Algebraic Properties of LimitsRead