Chapter 14: Q. 36 (page 883)

In Problems 7– 42, find each limit algebraically.
$\lim_{x \to 2} \frac{x^{3} - 8}{x^{2} - 4} .$

Short Answer

Expert verified

The answer is 3.

Step by step solution

Step. 1 Given Information

Firstly, we check whether the given function is in indeterminant form or not.

$f (x) = \frac{x^{3} - 8}{x^{2} - 4}$

Put $x = 2$ in the numerator we get,

$x^{3} - 8 = 2^{3} - 8 = 8 - 8 = 0$ .

Put $x = 2$ in the denominator we get,

$x^{2} - 4 = 2^{2} - 4 = 4 - 4 = 0$ .

Since both numerator and denominator gives 0 means they both have $x = 2$ as their common root.

Step. 2 Factorizing

Denominator, $x^{2} - 4 = (x + 2) (x - 2) .$

Numerator , $x^{3} - 8 = (x - 2) (x^{2} + 2 x + 4) .$

So,

$\lim_{x \to 2} \frac{x^{3} - 8}{x^{2} - 4} = \lim_{x \to 2} \frac{(x - 2) (x^{2} + 2 x + 4)}{(x - 2) (x + 2)} = \lim_{x \to 2} \frac{x^{2} + 2 x + 4}{x + 2} .$

Now we can put the limit directly.

Step. 3 Final calculation of the limit

$\lim_{x \to 2} \frac{x^{2} + 2 x + 4}{x + 2} = \frac{4 + 4 + 4}{2 + 2} = \frac{12}{4} = 3 .$

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In Problems 7– 42, find each limit algebraically.
$\lim_{x \to 2} \frac{x^{3} - 8}{x^{2} - 4} .$

Short Answer

Step by step solution

Step. 1 Given Information

Step. 2 Factorizing

Step. 3 Final calculation of the limit

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In Problems 7– 42, find each limit algebraically.limx→2x3-8x2-4.

Short Answer

Step by step solution

Step. 1 Given Information

Step. 2 Factorizing

Step. 3 Final calculation of the limit

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Discrete Mathematics

Mechanics Maths

Logic and Functions

Statistics

Study anywhere. Anytime. Across all devices.

In Problems 7– 42, find each limit algebraically.
$\lim_{x \to 2} \frac{x^{3} - 8}{x^{2} - 4} .$