Calculus Examples

$\frac{lim_{x \to - 8} 4.5 x^{4} + x^{3} - 2}{3 - 1.5 x^{4}}$

Step 1

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

$\frac{lim_{x \to - 8} 4.5 x^{4} + lim_{x \to - 8} x^{3} - lim_{x \to - 8} 2}{3 - 1.5 x^{4}}$

Step 2

Move the term $4.5$ outside of the limit because it is constant with respect to $x$ .

$\frac{4.5 lim_{x \to - 8} x^{4} + lim_{x \to - 8} x^{3} - lim_{x \to - 8} 2}{3 - 1.5 x^{4}}$

Step 3

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\frac{4.5 {(lim_{x \to - 8} x)}^{4} + lim_{x \to - 8} x^{3} - lim_{x \to - 8} 2}{3 - 1.5 x^{4}}$

Step 4

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

$\frac{4.5 {(lim_{x \to - 8} x)}^{4} + {(lim_{x \to - 8} x)}^{3} - lim_{x \to - 8} 2}{3 - 1.5 x^{4}}$

Step 5

Evaluate the limit of $2$ which is constant as $x$ approaches $- 8$ .

$\frac{4.5 {(lim_{x \to - 8} x)}^{4} + {(lim_{x \to - 8} x)}^{3} - 1 \cdot 2}{3 - 1.5 x^{4}}$

Step 6

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

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Step 6.1

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

$\frac{4.5 {(- 8)}^{4} + {(lim_{x \to - 8} x)}^{3} - 1 \cdot 2}{3 - 1.5 x^{4}}$

Step 6.2

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

$\frac{4.5 {(- 8)}^{4} + {(- 8)}^{3} - 1 \cdot 2}{3 - 1.5 x^{4}}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 8$ to the power of $4$ .

$\frac{4.5 \cdot 4096 + {(- 8)}^{3} - 1 \cdot 2}{3 - 1.5 x^{4}}$

Step 7.1.2

Multiply $4.5$ by $4096$ .

$\frac{18432 + {(- 8)}^{3} - 1 \cdot 2}{3 - 1.5 x^{4}}$

Step 7.1.3

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

$\frac{18432 - 512 - 1 \cdot 2}{3 - 1.5 x^{4}}$

Step 7.1.4

Multiply $- 1$ by $2$ .

$\frac{18432 - 512 - 2}{3 - 1.5 x^{4}}$

Step 7.1.5

Subtract $512$ from $18432$ .

$\frac{17920 - 2}{3 - 1.5 x^{4}}$

Step 7.1.6

Subtract $2$ from $17920$ .

$\frac{17918}{3 - 1.5 x^{4}}$

Step 7.2

Factor $1.5$ out of $3 - 1.5 x^{4}$ .

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Step 7.2.1

Factor $1.5$ out of $3$ .

$\frac{17918}{1.5 (2) - 1.5 x^{4}}$

Step 7.2.2

Factor $1.5$ out of $- 1.5 x^{4}$ .

$\frac{17918}{1.5 (2) + 1.5 (- x^{4})}$

Step 7.2.3

Factor $1.5$ out of $1.5 (2) + 1.5 (- x^{4})$ .

$\frac{17918}{1.5 (2 - x^{4})}$

Step 7.3

Factor $17918$ out of $17918$ .

$\frac{17918 (1)}{1.5 (2 - x^{4})}$

Step 7.4

Separate fractions.

$\frac{17918}{1.5} \cdot \frac{1}{2 - x^{4}}$

Step 7.5

Divide $17918$ by $1.5$ .

$11945.\overline{3} \frac{1}{2 - x^{4}}$

Step 7.6

Combine $11945.\overline{3}$ and $\frac{1}{2 - x^{4}}$ .

$\frac{11945.\overline{3}}{2 - x^{4}}$