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Evaluate the limit: lim_{{x o 3}} rac{x^3 - 27}{x - 3}.

Nov 17, 2023

Evaluate the limit: lim_{{x o 3}} rac{x^3 - 27}{x - 3}.

Here is a Step-by-Step solution: #step 1#

To evaluate the limit

\lim_{{x \to 3}} \frac{x^3 - 27}{x - 3}

, we first attempt to substitute

x=3

into the expression. However, this results in a

0/0

indeterminate form, which means we need to use algebraic manipulation to simplify the expression before taking the limit.

#step 2#

We recognize that the numerator

x^3 - 27

is a difference of cubes, which can be factored using the formula

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

. Here,

a = x

and

b = 3

#step 3#

Applying the difference of cubes formula to the numerator, we get:

x^3 - 27 = (x - 3)(x^2 + 3x + 9)

#step 4#

Now we can simplify the original expression by canceling out the common factor of

(x - 3)

from the numerator and the denominator:

\frac{x^3 - 27}{x - 3} = \frac{(x - 3)(x^2 + 3x + 9)}{x - 3} = x^2 + 3x + 9

#step 5#

With the expression simplified, we can now substitute

x = 3

directly into

x^2 + 3x + 9

to evaluate the limit:

\lim_{{x \to 3}} (x^2 + 3x + 9) = 3^2 + 3(3) + 9

#step 6#

Calculating the value, we get:

3^2 + 3(3) + 9 = 9 + 9 + 9 = 27

Answer

The limit is 27.

Key Concept

Factoring a difference of cubes to simplify a rational expression before taking a limit.

Explanation

The limit was evaluated by first recognizing the indeterminate form and then factoring the numerator as a difference of cubes. After canceling the common factor, the limit was found by direct substitution.