Calculus Examples

$lim_{x \to 4} (1 + \sqrt[3]{x}) (2 - x^{2} + 3 x^{3})$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $4$ .

$lim_{x \to 4} 1 + \sqrt[3]{x} \cdot lim_{x \to 4} 2 - x^{2} + 3 x^{3}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$(lim_{x \to 4} 1 + lim_{x \to 4} \sqrt[3]{x}) \cdot lim_{x \to 4} 2 - x^{2} + 3 x^{3}$

Step 3

Evaluate the limit of $1$ which is constant as $x$ approaches $4$ .

$(1 + lim_{x \to 4} \sqrt[3]{x}) \cdot lim_{x \to 4} 2 - x^{2} + 3 x^{3}$

Step 4

Move the limit under the radical sign.

$(1 + \sqrt[3]{lim_{x \to 4} x}) \cdot lim_{x \to 4} 2 - x^{2} + 3 x^{3}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$(1 + \sqrt[3]{lim_{x \to 4} x}) \cdot (lim_{x \to 4} 2 - lim_{x \to 4} x^{2} + lim_{x \to 4} 3 x^{3})$

Step 6

Evaluate the limit of $2$ which is constant as $x$ approaches $4$ .

$(1 + \sqrt[3]{lim_{x \to 4} x}) \cdot (2 - lim_{x \to 4} x^{2} + lim_{x \to 4} 3 x^{3})$

Step 7

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

$(1 + \sqrt[3]{lim_{x \to 4} x}) \cdot (2 - {(lim_{x \to 4} x)}^{2} + lim_{x \to 4} 3 x^{3})$

Step 8

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

$(1 + \sqrt[3]{lim_{x \to 4} x}) \cdot (2 - {(lim_{x \to 4} x)}^{2} + 3 lim_{x \to 4} x^{3})$

Step 9

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

$(1 + \sqrt[3]{lim_{x \to 4} x}) \cdot (2 - {(lim_{x \to 4} x)}^{2} + 3 {(lim_{x \to 4} x)}^{3})$

Step 10

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$(1 + \sqrt[3]{4}) \cdot (2 - {(lim_{x \to 4} x)}^{2} + 3 {(lim_{x \to 4} x)}^{3})$

Step 10.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$(1 + \sqrt[3]{4}) \cdot (2 - 4^{2} + 3 {(lim_{x \to 4} x)}^{3})$

Step 10.3

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$(1 + \sqrt[3]{4}) \cdot (2 - 4^{2} + 3 \cdot 4^{3})$

Step 11

Simplify the answer.

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Step 11.1

Simplify each term.

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Step 11.1.1

Raise $4$ to the power of $2$ .

$(1 + \sqrt[3]{4}) \cdot (2 - 1 \cdot 16 + 3 \cdot 4^{3})$

Step 11.1.2

Multiply $- 1$ by $16$ .

$(1 + \sqrt[3]{4}) \cdot (2 - 16 + 3 \cdot 4^{3})$

Step 11.1.3

Raise $4$ to the power of $3$ .

$(1 + \sqrt[3]{4}) \cdot (2 - 16 + 3 \cdot 64)$

Step 11.1.4

Multiply $3$ by $64$ .

$(1 + \sqrt[3]{4}) \cdot (2 - 16 + 192)$

Step 11.2

Subtract $16$ from $2$ .

$(1 + \sqrt[3]{4}) \cdot (- 14 + 192)$

Step 11.3

Add $- 14$ and $192$ .

$(1 + \sqrt[3]{4}) \cdot 178$

Step 11.4

Apply the distributive property.

$1 \cdot 178 + \sqrt[3]{4} \cdot 178$

Step 11.5

Multiply $178$ by $1$ .

$178 + \sqrt[3]{4} \cdot 178$

Step 11.6

Move $178$ to the left of $\sqrt[3]{4}$ .

$178 + 178 \sqrt[3]{4}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$178 + 178 \sqrt[3]{4}$

Decimal Form:

$460.55738725 \dots$