Calculus Examples

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

Step 1

Move the exponent $4$ from ${(3 x + 4 x^{2} - x^{3})}^{4}$ outside the limit using the Limits Power Rule.

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

Step 2

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

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

Step 3

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

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

Step 4

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

${(3 \cdot 0 + 4 \cdot 0^{2} - 0^{3})}^{4}$

Step 8

Simplify the answer.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $3$ by $0$ .

${(0 + 4 \cdot 0^{2} - 0^{3})}^{4}$

Step 8.1.2

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

${(0 + 4 \cdot 0 - 0^{3})}^{4}$

Step 8.1.3

Multiply $4$ by $0$ .

${(0 + 0 - 0^{3})}^{4}$

Step 8.1.4

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

${(0 + 0 - 0)}^{4}$

Step 8.1.5

Multiply $- 1$ by $0$ .

${(0 + 0 + 0)}^{4}$

Step 8.2

Add $0$ and $0$ .

${(0 + 0)}^{4}$

Step 8.3

Add $0$ and $0$ .

$0^{4}$

Step 8.4

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

$0$