Calculus Examples

$lim_{x \to - 8} \frac{1 - 3 x^{3}}{2 x^{3} - 6 x + 2}$

Step 1

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

$\frac{lim_{x \to - 8} 1 - 3 x^{3}}{lim_{x \to - 8} 2 x^{3} - 6 x + 2}$

Step 2

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

$\frac{lim_{x \to - 8} 1 - lim_{x \to - 8} 3 x^{3}}{lim_{x \to - 8} 2 x^{3} - 6 x + 2}$

Step 3

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

$\frac{1 - lim_{x \to - 8} 3 x^{3}}{lim_{x \to - 8} 2 x^{3} - 6 x + 2}$

Step 4

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

$\frac{1 - 3 lim_{x \to - 8} x^{3}}{lim_{x \to - 8} 2 x^{3} - 6 x + 2}$

Step 5

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

$\frac{1 - 3 {(lim_{x \to - 8} x)}^{3}}{lim_{x \to - 8} 2 x^{3} - 6 x + 2}$

Step 6

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

$\frac{1 - 3 {(lim_{x \to - 8} x)}^{3}}{lim_{x \to - 8} 2 x^{3} - lim_{x \to - 8} 6 x + lim_{x \to - 8} 2}$

Step 7

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

$\frac{1 - 3 {(lim_{x \to - 8} x)}^{3}}{2 lim_{x \to - 8} x^{3} - lim_{x \to - 8} 6 x + lim_{x \to - 8} 2}$

Step 8

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

$\frac{1 - 3 {(lim_{x \to - 8} x)}^{3}}{2 {(lim_{x \to - 8} x)}^{3} - lim_{x \to - 8} 6 x + lim_{x \to - 8} 2}$

Step 9

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

$\frac{1 - 3 {(lim_{x \to - 8} x)}^{3}}{2 {(lim_{x \to - 8} x)}^{3} - 6 lim_{x \to - 8} x + lim_{x \to - 8} 2}$

Step 10

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

$\frac{1 - 3 {(lim_{x \to - 8} x)}^{3}}{2 {(lim_{x \to - 8} x)}^{3} - 6 lim_{x \to - 8} x + 2}$

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 $x$ .

$\frac{1 - 3 {(- 8)}^{3}}{2 {(lim_{x \to - 8} x)}^{3} - 6 lim_{x \to - 8} x + 2}$

Step 11.2

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

$\frac{1 - 3 {(- 8)}^{3}}{2 {(- 8)}^{3} - 6 lim_{x \to - 8} x + 2}$

Step 11.3

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

$\frac{1 - 3 {(- 8)}^{3}}{2 {(- 8)}^{3} - 6 \cdot - 8 + 2}$

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

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

$\frac{1 - 3 \cdot - 512}{2 {(- 8)}^{3} - 6 \cdot - 8 + 2}$

Step 12.1.2

Multiply $- 3$ by $- 512$ .

$\frac{1 + 1536}{2 {(- 8)}^{3} - 6 \cdot - 8 + 2}$

Step 12.1.3

Add $1$ and $1536$ .

$\frac{1537}{2 {(- 8)}^{3} - 6 \cdot - 8 + 2}$

Step 12.2

Simplify the denominator.

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Step 12.2.1

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

$\frac{1537}{2 \cdot - 512 - 6 \cdot - 8 + 2}$

Step 12.2.2

Multiply $2$ by $- 512$ .

$\frac{1537}{- 1024 - 6 \cdot - 8 + 2}$

Step 12.2.3

Multiply $- 6$ by $- 8$ .

$\frac{1537}{- 1024 + 48 + 2}$

Step 12.2.4

Add $- 1024$ and $48$ .

$\frac{1537}{- 976 + 2}$

Step 12.2.5

Add $- 976$ and $2$ .

$\frac{1537}{- 974}$

Step 12.3

Move the negative in front of the fraction.

$- \frac{1537}{974}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$- \frac{1537}{974}$

Decimal Form:

$- 1.57802874 \dots$