Calculus Examples

$lim_{x \to 8} \frac{7 x^{3}}{x^{3} - 3 x^{2} + 6 x}$

Step 1

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$7 lim_{x \to 8} \frac{x^{3}}{x^{3} - 3 x^{2} + 6 x}$

Step 2

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

$7 \frac{lim_{x \to 8} x^{3}}{lim_{x \to 8} x^{3} - 3 x^{2} + 6 x}$

Step 3

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

$7 \frac{{(lim_{x \to 8} x)}^{3}}{lim_{x \to 8} x^{3} - 3 x^{2} + 6 x}$

Step 4

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

$7 \frac{{(lim_{x \to 8} x)}^{3}}{lim_{x \to 8} x^{3} - lim_{x \to 8} 3 x^{2} + lim_{x \to 8} 6 x}$

Step 5

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

$7 \frac{{(lim_{x \to 8} x)}^{3}}{{(lim_{x \to 8} x)}^{3} - lim_{x \to 8} 3 x^{2} + lim_{x \to 8} 6 x}$

Step 6

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

$7 \frac{{(lim_{x \to 8} x)}^{3}}{{(lim_{x \to 8} x)}^{3} - 3 lim_{x \to 8} x^{2} + lim_{x \to 8} 6 x}$

Step 7

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

$7 \frac{{(lim_{x \to 8} x)}^{3}}{{(lim_{x \to 8} x)}^{3} - 3 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 6 x}$

Step 8

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$7 \frac{{(lim_{x \to 8} x)}^{3}}{{(lim_{x \to 8} x)}^{3} - 3 {(lim_{x \to 8} x)}^{2} + 6 lim_{x \to 8} x}$

Step 9

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

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Step 9.1

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

$7 \frac{8^{3}}{{(lim_{x \to 8} x)}^{3} - 3 {(lim_{x \to 8} x)}^{2} + 6 lim_{x \to 8} x}$

Step 9.2

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

$7 \frac{8^{3}}{8^{3} - 3 {(lim_{x \to 8} x)}^{2} + 6 lim_{x \to 8} x}$

Step 9.3

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

$7 \frac{8^{3}}{8^{3} - 3 \cdot 8^{2} + 6 lim_{x \to 8} x}$

Step 9.4

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

$7 \frac{8^{3}}{8^{3} - 3 \cdot 8^{2} + 6 \cdot 8}$

Step 10

Simplify the answer.

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Step 10.1

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

$7 \frac{512}{8^{3} - 3 \cdot 8^{2} + 6 \cdot 8}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

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

$7 \frac{512}{512 - 3 \cdot 8^{2} + 6 \cdot 8}$

Step 10.2.2

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

$7 \frac{512}{512 - 3 \cdot 64 + 6 \cdot 8}$

Step 10.2.3

Multiply $- 3$ by $64$ .

$7 \frac{512}{512 - 192 + 6 \cdot 8}$

Step 10.2.4

Multiply $6$ by $8$ .

$7 \frac{512}{512 - 192 + 48}$

Step 10.2.5

Subtract $192$ from $512$ .

$7 \frac{512}{320 + 48}$

Step 10.2.6

Add $320$ and $48$ .

$7 (\frac{512}{368})$

Step 10.3

Cancel the common factor of $512$ and $368$ .

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Step 10.3.1

Factor $16$ out of $512$ .

$7 \frac{16 (32)}{368}$

Step 10.3.2

Cancel the common factors.

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Step 10.3.2.1

Factor $16$ out of $368$ .

$7 \frac{16 \cdot 32}{16 \cdot 23}$

Step 10.3.2.2

Cancel the common factor.

$7 \frac{16 \cdot 32}{16 \cdot 23}$

Step 10.3.2.3

Rewrite the expression.

$7 (\frac{32}{23})$

Step 10.4

Multiply $7 (\frac{32}{23})$ .

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Step 10.4.1

Combine $7$ and $\frac{32}{23}$ .

$\frac{7 \cdot 32}{23}$

Step 10.4.2

Multiply $7$ by $32$ .

$\frac{224}{23}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{224}{23}$

Decimal Form:

$9.73913043 \dots$