## 3.1 Limits: Numerical and Graphical Approaches

 This tutorial: Part A: Numerical Approach Next tutorial: Part B: Graphical Approach

(This topic appears in Section 3.1 in Applied Calculus or Section 10.1 in Finite Mathematics and Applied Calculus)

Estimating Limits Numerically

Look at the function

 f(x) = x3 - 8x - 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 - 8x - 2
11.41
11.9401
11.9940
11.9994
 2
 2.0001 2.001 2.01 2.1 12.0006 12.006 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:

 limx→2 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 11.994 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 x→2 g(x) = 12 The limit of g(x), as x approaches 2 from the left, equals 12 and lim x→2 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.

Definition of a Limit

 lim x→a f(x) = L As x approaches the number a from the left, f(x) approaches the number L lim x→a f(x) = R As x approaches the number a from the right, f(x) approaches the number R

If the left limit and the right limit exist and are equal (to L, say) then we say that limxaf(x) exists and equals L, and write

 limx→a f(x) = L. As x approaches the number a from both sides, f(x) approaches the single number L

 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-12x - 3
___
___
___
___
 3
 3.0001 3.001 3.01 3.1 ___ ___ ___ ___

 Q limx→3 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 numbers

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

 x -5.1 -5.01 -5.001 -5.0001 -5 -4.9999 -4.999 -4.99 -4.9 g(x) 23.2 23.1 23.001 23.0001 24 250000 24999.9 249.9 24.9

 Q lim  x→-5 g(x) = Select one 23 24 +infinity does not exist undefined 23.1 250,000 Q lim  x→-5 g(x) = Select one 23 24 +infinity does not exist undefined 23.1 250,000 Q lim x→-5 g(x) = Select one 23 24 +infinity does not exist undefined 23.1 250,000 Q g(-5) = Select one 23 24 +infinity does not exist undefined 23.1 250,000

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Last Updated: August, 2007