Calculus Examples

$lim_{x \to 0} \frac{x^{3} - 6 x + 8}{x - 2}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} x^{3} - 6 x + 8}{lim_{x \to 0} x - 2}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} x^{3} - lim_{x \to 0} 6 x + lim_{x \to 0} 8}{lim_{x \to 0} x - 2}$

Step 3

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

$\frac{{(lim_{x \to 0} x)}^{3} - lim_{x \to 0} 6 x + lim_{x \to 0} 8}{lim_{x \to 0} x - 2}$

Step 4

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

$\frac{{(lim_{x \to 0} x)}^{3} - 6 lim_{x \to 0} x + lim_{x \to 0} 8}{lim_{x \to 0} x - 2}$

Step 5

Evaluate the limit of $8$ which is constant as $x$ approaches $0$ .

$\frac{{(lim_{x \to 0} x)}^{3} - 6 lim_{x \to 0} x + 8}{lim_{x \to 0} x - 2}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{{(lim_{x \to 0} x)}^{3} - 6 lim_{x \to 0} x + 8}{lim_{x \to 0} x - lim_{x \to 0} 2}$

Step 7

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$\frac{{(lim_{x \to 0} x)}^{3} - 6 lim_{x \to 0} x + 8}{lim_{x \to 0} x - 1 \cdot 2}$

Step 8

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0^{3} - 6 lim_{x \to 0} x + 8}{lim_{x \to 0} x - 1 \cdot 2}$

Step 8.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0^{3} - 6 \cdot 0 + 8}{lim_{x \to 0} x - 1 \cdot 2}$

Step 8.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0^{3} - 6 \cdot 0 + 8}{0 - 1 \cdot 2}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{0 - 6 \cdot 0 + 8}{0 - 1 \cdot 2}$

Step 9.1.2

Multiply $- 6$ by $0$ .

$\frac{0 + 0 + 8}{0 - 1 \cdot 2}$

Step 9.1.3

Add $0$ and $0$ .

$\frac{0 + 8}{0 - 1 \cdot 2}$

Step 9.1.4

Add $0$ and $8$ .

$\frac{8}{0 - 1 \cdot 2}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $- 1$ by $2$ .

$\frac{8}{0 - 2}$

Step 9.2.2

Subtract $2$ from $0$ .

$\frac{8}{- 2}$

Step 9.3

Divide $8$ by $- 2$ .

$- 4$