Calculus Examples

$\frac{lim_{x \to 8} 3 x^{2} - 2 x}{9 x^{2} - 4}$

Step 1

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

$\frac{lim_{x \to 8} 3 x^{2} - lim_{x \to 8} 2 x}{9 x^{2} - 4}$

Step 2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 lim_{x \to 8} x^{2} - lim_{x \to 8} 2 x}{9 x^{2} - 4}$

Step 3

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{3 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 2 x}{9 x^{2} - 4}$

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 {(lim_{x \to 8} x)}^{2} - 2 lim_{x \to 8} x}{9 x^{2} - 4}$

Step 5

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

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Step 5.1

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

$\frac{3 \cdot 8^{2} - 2 lim_{x \to 8} x}{9 x^{2} - 4}$

Step 5.2

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

$\frac{3 \cdot 8^{2} - 2 \cdot 8}{9 x^{2} - 4}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $8$ to the power of $2$ .

$\frac{3 \cdot 64 - 2 \cdot 8}{9 x^{2} - 4}$

Step 6.1.2

Multiply $3$ by $64$ .

$\frac{192 - 2 \cdot 8}{9 x^{2} - 4}$

Step 6.1.3

Multiply $- 2$ by $8$ .

$\frac{192 - 16}{9 x^{2} - 4}$

Step 6.1.4

Subtract $16$ from $192$ .

$\frac{176}{9 x^{2} - 4}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$\frac{176}{{(3 x)}^{2} - 4}$

Step 6.2.2

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

$\frac{176}{{(3 x)}^{2} - 2^{2}}$

Step 6.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 x$ and $b = 2$ .

$\frac{176}{(3 x + 2) (3 x - 2)}$