# The limit of the function lim x → − 3 x 2 + 3 x x 2 − x − 12 .

### 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 2.3, Problem 10E
To determine

## To evaluate: The limit of the function limx→−3x2+3xx2−x−12.

Expert Solution

The limit of the function is 37_.

### Explanation of Solution

Limit Laws:

Suppose that c is a constant and the limits limxaf(x) and limxag(x) exist, then

Limit law 2:limxa[f(x)g(x)]=limxaf(x)limxag(x)

Limit law 7:limxac=c

Limit law 8:limxax=a

Direct substitution property:

If f is a polynomial or a rational function and a is in the domain of f, then limxaf(x)=f(a) .

Fact 1:

If f(x)=g(x) when xa, then limxaf(x)=limxag(x) , provided the limit exist.

Calculation:

Compute the limit value of the denominator.

limx3(x2x12)=limx3(x2)limx3(x)limx3(12) (by limit law 2)=(3)2(3)12 (by limit law 9,8 and 7)=9+312 (by limit law 8)=0

Since the limit of the denominator is zero, the quotient law cannot be applied.

Let f(x)=x2+3xx2x12 (1)

Simplify f(x) by using elementary algebra.

Factorize the numerator x2x12,

x2x12=x24x+3x12=x(x4)+3(x4)=(x4)(x+3)

Substitute (x4)(x+3) for x2x12 in equation (1).

f(x)=x2+3x(x4)(x+3)

Take the common terms out from the numerator, f(x)=x(x+3)(x4)(x+3).

Cancel the common terms from both the numerator and the denominator, f(x)=x(x4).

Use fact 1, f(x)=x(x4) and x3, then limx3x2+3xx2x12=limx3x(x4).

Apply direct substitution property on the limit function.

limx3x(x4)=3(34)=37=37

Thus, the limit of the function is 37_.

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