CK-12-Pre-Calculus Concepts

14.3. Tables to Find Limits http://www.ck12.org

limx→ 0

√x+ 3 −√ 3

x

TABLE14.6:

x -0.1 -0.01 -0.001 0.001 0.01 0.1

f(x)

Solution:

TABLE14.7:

x -0.1 -0.01 -0.001 0.001 0.01 0.1

f(x) 0.29112 0.28892 0.2887 0.28865 0.28843 0.28631

The evidence suggests that the limit is a number between 0.2887 and 0.28865. When you learn to find the limit
analytically, you will know that the exact limit is^12 · 312 ≈ 0 .2886751346.
Concept Problem Revisited
When you enter values close to -1 in the table you getyvalues that are increasingly close to the number 1. This
implies that the limit asxapproaches -1 is 1. Notice that when you evaluate the function at -1, the calculator produces
an error. This should lead you to the conclusion that while the function is not defined atx=−1 , the limit does exist.

Vocabulary

Numericallyis a term used to describe one of several different representations in mathematics. It refers to tables
where the actual numbers are visible.

Guided Practice

1. Graph the following function and the use a table to verify the limit asxapproaches 1.
   f(x) =xx^3 −− 11 ,x 6 = 1


2. Estimate the limit numerically.
   limx→ 2 x^2 −x−^3 x 2 +^2


3. Estimate the limit numerically.

limx→ 0 [
x+^42 ]−^2
x
Answers:

1. limx→ 1 f(x) =3. This is because when you factor the numerator and cancel common factors, the function becomes
   a quadratic with a hole at the point( 1 , 3 ).