Calculus Examples

$\frac{lim_{x \to 0} x^{3} + 12 x^{2} - 5 x}{5 x}$

Step 1

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

$\frac{lim_{x \to 0} x^{3} + lim_{x \to 0} 12 x^{2} - lim_{x \to 0} 5 x}{5 x}$

Step 2

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

$\frac{{(lim_{x \to 0} x)}^{3} + lim_{x \to 0} 12 x^{2} - lim_{x \to 0} 5 x}{5 x}$

Step 3

Move the term $12$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 0} x)}^{3} + 12 lim_{x \to 0} x^{2} - lim_{x \to 0} 5 x}{5 x}$

Step 4

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

$\frac{{(lim_{x \to 0} x)}^{3} + 12 {(lim_{x \to 0} x)}^{2} - lim_{x \to 0} 5 x}{5 x}$

Step 5

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

$\frac{{(lim_{x \to 0} x)}^{3} + 12 {(lim_{x \to 0} x)}^{2} - 5 lim_{x \to 0} x}{5 x}$

Step 6

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

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Step 6.1

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

$\frac{0^{3} + 12 {(lim_{x \to 0} x)}^{2} - 5 lim_{x \to 0} x}{5 x}$

Step 6.2

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

$\frac{0^{3} + 12 \cdot 0^{2} - 5 lim_{x \to 0} x}{5 x}$

Step 6.3

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

$\frac{0^{3} + 12 \cdot 0^{2} - 5 \cdot 0}{5 x}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$\frac{0 + 12 \cdot 0^{2} - 5 \cdot 0}{5 x}$

Step 7.1.2

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

$\frac{0 + 12 \cdot 0 - 5 \cdot 0}{5 x}$

Step 7.1.3

Multiply $12$ by $0$ .

$\frac{0 + 0 - 5 \cdot 0}{5 x}$

Step 7.1.4

Multiply $- 5$ by $0$ .

$\frac{0 + 0 + 0}{5 x}$

Step 7.1.5

Add $0 + 0$ and $0$ .

$\frac{0 + 0}{5 x}$

Step 7.1.6

Add $0$ and $0$ .

$\frac{0}{5 x}$

Step 7.2

Cancel the common factor of $0$ and $5$ .

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Step 7.2.1

Factor $5$ out of $0$ .

$\frac{5 (0)}{5 x}$

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

Factor $5$ out of $5 x$ .

$\frac{5 (0)}{5 (x)}$

Step 7.2.2.2

Cancel the common factor.

$\frac{5 \cdot 0}{5 x}$

Step 7.2.2.3

Rewrite the expression.

$\frac{0}{x}$

Step 7.3

Divide $0$ by $x$ .

$0$