3.1 Limits: Numerical and Graphical Approaches

Estimating Limits Numerically
(Similar to Example 1(a) in Section 3.1 of Applied Calculus and Section 10.1 in Finite Mathematics and Applied Calculus)

Look at the function

f(x)

=

x3 - 8 x - 2

and ask yourself: "What happens to f(x) as x approaches 2?" (Notice that you cannot simply substitute x = 2, because the function is not defined at x = 2.) The following chart shows the value of f(x) for values of x close to, and on either side of 2:

x approaching 2 from the left		x approaching 2 from the right
x 1.9 1.99 1.999 1.9999 f(x) = x3 - 8 x - 2 11.41 11.9401 11.9940 11.9994	2	2.0001 2.001 2.01 2.1 12.0006 12.0060 12.0601 12.61

We have left the entry under 2 blank to emphasize that, when calculating the limit of f(x) as x approaches 2, we are not interested in its value when x equals 2. Notice from the table that, the closer x gets to 2 from either side, the closer f(x) gets to 12. We write this as:

lim x2

f(x) = 12

In words:

The limit of f(x), as x approaches 2, equals 12.

Q What if we had gotten different answers when approaching 2 from the left and right?
A Suppose, for instance, that the table looked like this:

x approaching 2 from the left		x approaching 2 from the right
x 1.9 1.99 1.999 1.9999 g(x) 11.41 11.9401 11.9940 11.9994	2	2.0001 2.001 2.01 2.1 4.3333 4.3301 4.3024 4.1039

Notice that the limit appears to be 12 as you approach from the left, but it now appears to be 4¹/₃ if you approach from the right. We therefore write:

lim  x2	g(x) = 12		The limit of g(x), as x approaches 2 from the left, equals 12
and
lim  x2	g(x) = 41/3		The limit of g(x), as x approaches 2 from the right, equals 41/3

Before going on to the first practice question, look over the following summary of terms.

First calculate the missing values in the following table (might we suggest you use the Function Evaluator & Grapher for this) and then decide on a numerical estimate of limx → 3f(x).

x approaching 3 from the left		x approaching 3 from the right
x 2.9 2.99 2.999 2.9999 f(x) = x2+x-12 x - 3 ___ ___ ___ ___	3	3.0001 3.001 3.01 3.1 ___ ___ ___ ___

Q

lim x3

f(x) = ?

	7.0			5.0			-3.0
	There is no limit; the numbers on the left are big positive numbers, while those on the right are big negative numers

Suppose a certain function g has the following table of values.

Now try the exercises in Section 3.1 in Applied Calculus or Section 10.1 in Finite Mathematics and Applied Calculus. Don't try the ones based on graphs until you have looked at Part B: Graphical Approach.