Calculus Examples

$f (x) = \frac{x^{2} - 4}{(x^{2} - 3 x - 10) lim_{x \to 2} f (x)}$

Step 1

Evaluate the limit of $f (x)$ by plugging in $2$ for $x$ .

$f (x) = \frac{x^{2} - 4}{(x^{2} - 3 x - 10) f (2)}$

Step 2

Simplify the answer.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Rewrite $4$ as $2^{2}$ .

$f (x) = \frac{x^{2} - 2^{2}}{(x^{2} - 3 x - 10) f (2)}$

Step 2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$f (x) = \frac{(x + 2) (x - 2)}{(x^{2} - 3 x - 10) f (2)}$

Step 2.2

Factor $x^{2} - 3 x - 10$ using the AC method.

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Step 2.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Step 2.2.2

Write the factored form using these integers.

$f (x) = \frac{(x + 2) (x - 2)}{(x - 5) (x + 2) f (2)}$

Step 2.3

Cancel the common factor of $x + 2$ .

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Step 2.3.1

Cancel the common factor.

$f (x) = \frac{(x + 2) (x - 2)}{(x - 5) (x + 2) f (2)}$

Step 2.3.2

Rewrite the expression.

$f (x) = \frac{x - 2}{(x - 5) f (2)}$

Step 2.4

Reorder factors in $f (x) = \frac{x - 2}{(x - 5) f (2)}$ .

$f (x) = \frac{x - 2}{f (2) (x - 5)}$