Calculus Examples

$lim_{x \to 8} 15 x^{3} - \frac{4}{5 x^{2} + 16 x}$

Step 1

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

$lim_{x \to 8} 15 x^{3} - lim_{x \to 8} \frac{4}{5 x^{2} + 16 x}$

Step 2

Move the term $15$ outside of the limit because it is constant with respect to $x$ .

$15 lim_{x \to 8} x^{3} - lim_{x \to 8} \frac{4}{5 x^{2} + 16 x}$

Step 3

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$15 {(lim_{x \to 8} x)}^{3} - lim_{x \to 8} \frac{4}{5 x^{2} + 16 x}$

Step 4

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$15 {(lim_{x \to 8} x)}^{3} - 4 lim_{x \to 8} \frac{1}{5 x^{2} + 16 x}$

Step 5

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

$15 {(lim_{x \to 8} x)}^{3} - 4 \frac{lim_{x \to 8} 1}{lim_{x \to 8} 5 x^{2} + 16 x}$

Step 6

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$15 {(lim_{x \to 8} x)}^{3} - 4 \frac{1}{lim_{x \to 8} 5 x^{2} + 16 x}$

Step 7

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

$15 {(lim_{x \to 8} x)}^{3} - 4 \frac{1}{lim_{x \to 8} 5 x^{2} + lim_{x \to 8} 16 x}$

Step 8

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$15 {(lim_{x \to 8} x)}^{3} - 4 \frac{1}{5 lim_{x \to 8} x^{2} + lim_{x \to 8} 16 x}$

Step 9

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

$15 {(lim_{x \to 8} x)}^{3} - 4 \frac{1}{5 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 16 x}$

Step 10

Move the term $16$ outside of the limit because it is constant with respect to $x$ .

$15 {(lim_{x \to 8} x)}^{3} - 4 \frac{1}{5 {(lim_{x \to 8} x)}^{2} + 16 lim_{x \to 8} x}$

Step 11

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

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Step 11.1

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

$15 \cdot 8^{3} - 4 \frac{1}{5 {(lim_{x \to 8} x)}^{2} + 16 lim_{x \to 8} x}$

Step 11.2

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

$15 \cdot 8^{3} - 4 \frac{1}{5 \cdot 8^{2} + 16 lim_{x \to 8} x}$

Step 11.3

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

$15 \cdot 8^{3} - 4 \frac{1}{5 \cdot 8^{2} + 16 \cdot 8}$

Step 12

Simplify the answer.

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Step 12.1

Simplify each term.

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Step 12.1.1

Raise $8$ to the power of $3$ .

$15 \cdot 512 - 4 \frac{1}{5 \cdot 8^{2} + 16 \cdot 8}$

Step 12.1.2

Multiply $15$ by $512$ .

$7680 - 4 \frac{1}{5 \cdot 8^{2} + 16 \cdot 8}$

Step 12.1.3

Simplify the denominator.

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Step 12.1.3.1

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

$7680 - 4 \frac{1}{5 \cdot 64 + 16 \cdot 8}$

Step 12.1.3.2

Multiply $5$ by $64$ .

$7680 - 4 \frac{1}{320 + 16 \cdot 8}$

Step 12.1.3.3

Multiply $16$ by $8$ .

$7680 - 4 \frac{1}{320 + 128}$

Step 12.1.3.4

Add $320$ and $128$ .

$7680 - 4 (\frac{1}{448})$

Step 12.1.4

Cancel the common factor of $4$ .

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Step 12.1.4.1

Factor $4$ out of $- 4$ .

$7680 + 4 (- 1) \frac{1}{448}$

Step 12.1.4.2

Factor $4$ out of $448$ .

$7680 + 4 \cdot - 1 \frac{1}{4 \cdot 112}$

Step 12.1.4.3

Cancel the common factor.

$7680 + 4 \cdot - 1 \frac{1}{4 \cdot 112}$

Step 12.1.4.4

Rewrite the expression.

$7680 - \frac{1}{112}$

Step 12.2

To write $7680$ as a fraction with a common denominator, multiply by $\frac{112}{112}$ .

$7680 \cdot \frac{112}{112} - \frac{1}{112}$

Step 12.3

Combine $7680$ and $\frac{112}{112}$ .

$\frac{7680 \cdot 112}{112} - \frac{1}{112}$

Step 12.4

Combine the numerators over the common denominator.

$\frac{7680 \cdot 112 - 1}{112}$

Step 12.5

Simplify the numerator.

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Step 12.5.1

Multiply $7680$ by $112$ .

$\frac{860160 - 1}{112}$

Step 12.5.2

Subtract $1$ from $860160$ .

$\frac{860159}{112}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{860159}{112}$

Decimal Form:

$7679.99107142 \dots$