Calculus Examples

$\frac{lim_{x \to 8} 5 x^{3} + 2 x}{- 3 x^{4} + 3 x^{3} + 6}$

Step 1

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

$\frac{lim_{x \to 8} 5 x^{3} + lim_{x \to 8} 2 x}{- 3 x^{4} + 3 x^{3} + 6}$

Step 2

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

$\frac{5 lim_{x \to 8} x^{3} + lim_{x \to 8} 2 x}{- 3 x^{4} + 3 x^{3} + 6}$

Step 3

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

$\frac{5 {(lim_{x \to 8} x)}^{3} + lim_{x \to 8} 2 x}{- 3 x^{4} + 3 x^{3} + 6}$

Step 4

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

$\frac{5 {(lim_{x \to 8} x)}^{3} + 2 lim_{x \to 8} x}{- 3 x^{4} + 3 x^{3} + 6}$

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 $x$ .

$\frac{5 \cdot 8^{3} + 2 lim_{x \to 8} x}{- 3 x^{4} + 3 x^{3} + 6}$

Step 5.2

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

$\frac{5 \cdot 8^{3} + 2 \cdot 8}{- 3 x^{4} + 3 x^{3} + 6}$

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 $3$ .

$\frac{5 \cdot 512 + 2 \cdot 8}{- 3 x^{4} + 3 x^{3} + 6}$

Step 6.1.2

Multiply $5$ by $512$ .

$\frac{2560 + 2 \cdot 8}{- 3 x^{4} + 3 x^{3} + 6}$

Step 6.1.3

Multiply $2$ by $8$ .

$\frac{2560 + 16}{- 3 x^{4} + 3 x^{3} + 6}$

Step 6.1.4

Add $2560$ and $16$ .

$\frac{2576}{- 3 x^{4} + 3 x^{3} + 6}$

Step 6.2

Factor $3$ out of $- 3 x^{4} + 3 x^{3} + 6$ .

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Step 6.2.1

Factor $3$ out of $- 3 x^{4}$ .

$\frac{2576}{3 (- x^{4}) + 3 x^{3} + 6}$

Step 6.2.2

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

$\frac{2576}{3 (- x^{4}) + 3 (x^{3}) + 6}$

Step 6.2.3

Factor $3$ out of $6$ .

$\frac{2576}{3 (- x^{4}) + 3 x^{3} + 3 \cdot 2}$

Step 6.2.4

Factor $3$ out of $3 (- x^{4}) + 3 x^{3}$ .

$\frac{2576}{3 (- x^{4} + x^{3}) + 3 \cdot 2}$

Step 6.2.5

Factor $3$ out of $3 (- x^{4} + x^{3}) + 3 \cdot 2$ .

$\frac{2576}{3 (- x^{4} + x^{3} + 2)}$

Step 6.3

Factor $- 1$ out of $- x^{4}$ .

$\frac{2576}{3 (- (x^{4}) + x^{3} + 2)}$

Step 6.4

Factor $- 1$ out of $x^{3}$ .

$\frac{2576}{3 (- (x^{4}) - 1 (- x^{3}) + 2)}$

Step 6.5

Factor $- 1$ out of $- (x^{4}) - 1 (- x^{3})$ .

$\frac{2576}{3 (- (x^{4} - x^{3}) + 2)}$

Step 6.6

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{2576}{3 (- (x^{4} - x^{3}) - 1 (- 2))}$

Step 6.7

Factor $- 1$ out of $- (x^{4} - x^{3}) - 1 (- 2)$ .

$\frac{2576}{3 (- (x^{4} - x^{3} - 2))}$

Step 6.8

Rewrite $- (x^{4} - x^{3} - 2)$ as $- 1 (x^{4} - x^{3} - 2)$ .

$\frac{2576}{3 (- 1 (x^{4} - x^{3} - 2))}$

Step 6.9

Move the negative in front of the fraction.

$- \frac{2576}{3 (x^{4} - x^{3} - 2)}$