Calculus Examples

$\frac{lim_{x \to 8} (x) (x - 4)}{3 x^{2} - 8 x - 16} + 4$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} x \cdot lim_{x \to 8} x - 4}{3 x^{2} - 8 x - 16} + 4$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} x \cdot (lim_{x \to 8} x - lim_{x \to 8} 4)}{3 x^{2} - 8 x - 16} + 4$

Step 3

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

$\frac{lim_{x \to 8} x \cdot (lim_{x \to 8} x - 1 \cdot 4)}{3 x^{2} - 8 x - 16} + 4$

Step 4

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 4.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{8 \cdot (lim_{x \to 8} x - 1 \cdot 4)}{3 x^{2} - 8 x - 16} + 4$

Step 4.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{8 \cdot (8 - 1 \cdot 4)}{3 x^{2} - 8 x - 16} + 4$

Step 5

Simplify the answer.

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Step 5.1

Multiply $- 1$ by $4$ .

$\frac{8 \cdot (8 - 4)}{3 x^{2} - 8 x - 16} + 4$

Step 5.2

Subtract $4$ from $8$ .

$\frac{8 \cdot 4}{3 x^{2} - 8 x - 16} + 4$

Step 5.3

Multiply $8$ by $4$ .

$\frac{32}{3 x^{2} - 8 x - 16} + 4$

Step 5.4

Factor by grouping.

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Step 5.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 16 = - 48$ and whose sum is $b = - 8$ .

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Step 5.4.1.1

Factor $- 8$ out of $- 8 x$ .

$\frac{32}{3 x^{2} - 8 (x) - 16} + 4$

Step 5.4.1.2

Rewrite $- 8$ as $4$ plus $- 12$

$\frac{32}{3 x^{2} + (4 - 12) x - 16} + 4$

Step 5.4.1.3

Apply the distributive property.

$\frac{32}{3 x^{2} + 4 x - 12 x - 16} + 4$

Step 5.4.2

Factor out the greatest common factor from each group.

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Step 5.4.2.1

Group the first two terms and the last two terms.

$\frac{32}{(3 x^{2} + 4 x) - 12 x - 16} + 4$

Step 5.4.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{32}{x (3 x + 4) - 4 (3 x + 4)} + 4$

Step 5.4.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 4$ .

$\frac{32}{(3 x + 4) (x - 4)} + 4$