Calculus Examples

$\frac{lim_{x \to 0} 5 x^{4} + 27 x^{2}}{3 x^{2} - 6 x^{2}}$

Step 1

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

$\frac{lim_{x \to 0} 5 x^{4} + lim_{x \to 0} 27 x^{2}}{3 x^{2} - 6 x^{2}}$

Step 2

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

$\frac{5 lim_{x \to 0} x^{4} + lim_{x \to 0} 27 x^{2}}{3 x^{2} - 6 x^{2}}$

Step 3

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

$\frac{5 {(lim_{x \to 0} x)}^{4} + lim_{x \to 0} 27 x^{2}}{3 x^{2} - 6 x^{2}}$

Step 4

Move the term $27$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 {(lim_{x \to 0} x)}^{4} + 27 lim_{x \to 0} x^{2}}{3 x^{2} - 6 x^{2}}$

Step 5

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

$\frac{5 {(lim_{x \to 0} x)}^{4} + 27 {(lim_{x \to 0} x)}^{2}}{3 x^{2} - 6 x^{2}}$

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 $x$ .

$\frac{5 \cdot 0^{4} + 27 {(lim_{x \to 0} x)}^{2}}{3 x^{2} - 6 x^{2}}$

Step 6.2

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

$\frac{5 \cdot 0^{4} + 27 \cdot 0^{2}}{3 x^{2} - 6 x^{2}}$

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 $0$ .

$\frac{5 \cdot 0 + 27 \cdot 0^{2}}{3 x^{2} - 6 x^{2}}$

Step 7.1.2

Multiply $5$ by $0$ .

$\frac{0 + 27 \cdot 0^{2}}{3 x^{2} - 6 x^{2}}$

Step 7.1.3

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

$\frac{0 + 27 \cdot 0}{3 x^{2} - 6 x^{2}}$

Step 7.1.4

Multiply $27$ by $0$ .

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

Step 7.1.5

Add $0$ and $0$ .

$\frac{0}{3 x^{2} - 6 x^{2}}$

Step 7.2

Subtract $6 x^{2}$ from $3 x^{2}$ .

$\frac{0}{- 3 x^{2}}$

Step 7.3

Cancel the common factor of $0$ and $- 3$ .

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Step 7.3.1

Factor $3$ out of $0$ .

$\frac{3 (0)}{- 3 x^{2}}$

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

Factor $3$ out of $- 3 x^{2}$ .

$\frac{3 (0)}{3 (- x^{2})}$

Step 7.3.2.2

Cancel the common factor.

$\frac{3 \cdot 0}{3 (- x^{2})}$

Step 7.3.2.3

Rewrite the expression.

$\frac{0}{- x^{2}}$

Step 7.4

Divide $0$ by $- x^{2}$ .

$0$