Calculus Examples

$\frac{lim_{x \to 0} 5 x^{3} + 8 x^{2}}{3 x^{4} - 16 x^{2}}$

Step 1

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

$\frac{lim_{x \to 0} 5 x^{3} + lim_{x \to 0} 8 x^{2}}{3 x^{4} - 16 x^{2}}$

Step 2

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

$\frac{5 lim_{x \to 0} x^{3} + lim_{x \to 0} 8 x^{2}}{3 x^{4} - 16 x^{2}}$

Step 3

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

$\frac{5 {(lim_{x \to 0} x)}^{3} + lim_{x \to 0} 8 x^{2}}{3 x^{4} - 16 x^{2}}$

Step 4

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 {(lim_{x \to 0} x)}^{3} + 8 lim_{x \to 0} x^{2}}{3 x^{4} - 16 x^{2}}$

Step 5

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

$\frac{5 {(lim_{x \to 0} x)}^{3} + 8 {(lim_{x \to 0} x)}^{2}}{3 x^{4} - 16 x^{2}}$

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 $x$ .

$\frac{5 \cdot 0^{3} + 8 {(lim_{x \to 0} x)}^{2}}{3 x^{4} - 16 x^{2}}$

Step 6.2

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

$\frac{5 \cdot 0^{3} + 8 \cdot 0^{2}}{3 x^{4} - 16 x^{2}}$

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 $0$ .

$\frac{5 \cdot 0 + 8 \cdot 0^{2}}{3 x^{4} - 16 x^{2}}$

Step 7.1.2

Multiply $5$ by $0$ .

$\frac{0 + 8 \cdot 0^{2}}{3 x^{4} - 16 x^{2}}$

Step 7.1.3

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

$\frac{0 + 8 \cdot 0}{3 x^{4} - 16 x^{2}}$

Step 7.1.4

Multiply $8$ by $0$ .

$\frac{0 + 0}{3 x^{4} - 16 x^{2}}$

Step 7.1.5

Add $0$ and $0$ .

$\frac{0}{3 x^{4} - 16 x^{2}}$

Step 7.2

Factor $x^{2}$ out of $3 x^{4} - 16 x^{2}$ .

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Step 7.2.1

Factor $x^{2}$ out of $3 x^{4}$ .

$\frac{0}{x^{2} (3 x^{2}) - 16 x^{2}}$

Step 7.2.2

Factor $x^{2}$ out of $- 16 x^{2}$ .

$\frac{0}{x^{2} (3 x^{2}) + x^{2} \cdot - 16}$

Step 7.2.3

Factor $x^{2}$ out of $x^{2} (3 x^{2}) + x^{2} \cdot - 16$ .

$\frac{0}{x^{2} (3 x^{2} - 16)}$

Step 7.3

Divide $0$ by $x^{2} (3 x^{2} - 16)$ .

$0$