# To guess: The limit of the function lim x → 3 x 2 − 3 x x 2 − 9 . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 2.2, Problem 17E
To determine

## To guess: The limit of the function limx→3x2−3xx2−9.

Expert Solution

The limit of the function is guessed to 0.5.

### Explanation of Solution

Given:

The values of x are 3.1,3.05,3.01,3.001,3.0001,2.9,2.95,2.99,2.999 and 2.9999.

Calculation:

Since x approaches 3, the values 3.1,3.05,3.01,3.001, and 3.0001 are to the right of 3 and the values 2.9,2.95,2.99,2.999 and 2.9999 are to the left of 3.

Evaluate the function (correct to 6 decimal places) when x=3.1,3.05,3.01,3.001, and 3.0001 as shown in the below table.

 x x2−3x x2−9 f(x)=x2−3xx2−9 3.1 0.31 0.61 0.508197 3.05 0.1525 0.3025 0.504132 3.01 0.0301 0.0601 0.500832 3.001 0.003001 0.006001 0.500083 3.0001 0.0003 0.0006 0.500008

Here, the value of f(x) gets closer to the value 0.5 as x approaches to 3 from the right side.

That is, limx3+x23xx29=0.5.

Evaluate the function (correct to 6 decimal places) when x=2.9,2.95,2.99,2.999 and 2.9999 as shown in the below table.

 x x2−3x x2−9 f(x)=x2−3xx2−9 2.9 −0.29 −0.59 0.491525 2.95 −0.1475 −0.2975 0.495798 2.99 −0.0299 −0.0599 0.499165 2.999 −0.003 −0.006 0.499917 2.9999 −0.0003 −0.0006 0.499992

Here, the value of f(x) gets closer to the value 0.5 as x approaches to 3 from the left side.

That is, limx3x23xx29=0.5.

Since the right hand limit and the left hand limits are the same, limx3x23xx29 exists.

Therefore, the limit of the function is guessed to 0.5.

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