Calculus Examples

$\frac{lim_{x \to 8} x^{2} - 3 x - 28}{2 x^{3} - 98 x}$

Step 1

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

$\frac{lim_{x \to 8} x^{2} - lim_{x \to 8} 3 x - lim_{x \to 8} 28}{2 x^{3} - 98 x}$

Step 2

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

$\frac{{(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 3 x - lim_{x \to 8} 28}{2 x^{3} - 98 x}$

Step 3

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

$\frac{{(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x - lim_{x \to 8} 28}{2 x^{3} - 98 x}$

Step 4

Evaluate the limit of $28$ which is constant as $x$ approaches $8$ .

$\frac{{(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x - 1 \cdot 28}{2 x^{3} - 98 x}$

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 $x$ .

$\frac{8^{2} - 3 lim_{x \to 8} x - 1 \cdot 28}{2 x^{3} - 98 x}$

Step 5.2

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

$\frac{8^{2} - 3 \cdot 8 - 1 \cdot 28}{2 x^{3} - 98 x}$

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 $2$ .

$\frac{64 - 3 \cdot 8 - 1 \cdot 28}{2 x^{3} - 98 x}$

Step 6.1.2

Multiply $- 3$ by $8$ .

$\frac{64 - 24 - 1 \cdot 28}{2 x^{3} - 98 x}$

Step 6.1.3

Multiply $- 1$ by $28$ .

$\frac{64 - 24 - 28}{2 x^{3} - 98 x}$

Step 6.1.4

Subtract $24$ from $64$ .

$\frac{40 - 28}{2 x^{3} - 98 x}$

Step 6.1.5

Subtract $28$ from $40$ .

$\frac{12}{2 x^{3} - 98 x}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Factor $2 x$ out of $2 x^{3} - 98 x$ .

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Step 6.2.1.1

Factor $2 x$ out of $2 x^{3}$ .

$\frac{12}{2 x (x^{2}) - 98 x}$

Step 6.2.1.2

Factor $2 x$ out of $- 98 x$ .

$\frac{12}{2 x (x^{2}) + 2 x (- 49)}$

Step 6.2.1.3

Factor $2 x$ out of $2 x (x^{2}) + 2 x (- 49)$ .

$\frac{12}{2 x (x^{2} - 49)}$

Step 6.2.2

Rewrite $49$ as $7^{2}$ .

$\frac{12}{2 x (x^{2} - 7^{2})}$

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 $a = x$ and $b = 7$ .

$\frac{12}{2 x (x + 7) (x - 7)}$

Step 6.3

Cancel the common factor of $12$ and $2$ .

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Step 6.3.1

Factor $2$ out of $12$ .

$\frac{2 \cdot 6}{2 x (x + 7) (x - 7)}$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $2$ out of $2 x (x + 7) (x - 7)$ .

$\frac{2 \cdot 6}{2 ((x (x + 7)) (x - 7))}$

Step 6.3.2.2

Cancel the common factor.

$\frac{2 \cdot 6}{2 ((x (x + 7)) (x - 7))}$

Step 6.3.2.3

Rewrite the expression.

$\frac{6}{(x (x + 7)) (x - 7)}$

$\frac{6}{x (x + 7) (x - 7)}$