Calculus Examples

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

Step 1

Split the limit using the Product of Limits Rule on the limit as $k$ approaches $2$ .

$lim_{k \to 2} \sqrt{3 k^{2} + 4} \cdot lim_{k \to 2} \sqrt[3]{3 k + 2}$

Step 2

Move the limit under the radical sign.

$\sqrt{lim_{k \to 2} 3 k^{2} + 4} \cdot lim_{k \to 2} \sqrt[3]{3 k + 2}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $k$ approaches $2$ .

$\sqrt{lim_{k \to 2} 3 k^{2} + lim_{k \to 2} 4} \cdot lim_{k \to 2} \sqrt[3]{3 k + 2}$

Step 4

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

$\sqrt{3 lim_{k \to 2} k^{2} + lim_{k \to 2} 4} \cdot lim_{k \to 2} \sqrt[3]{3 k + 2}$

Step 5

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

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

Step 6

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

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

Step 7

Move the limit under the radical sign.

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

Step 8

Split the limit using the Sum of Limits Rule on the limit as $k$ approaches $2$ .

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

Step 9

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

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

Step 10

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

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

Step 11

Evaluate the limits by plugging in $2$ for all occurrences of $k$ .

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Step 11.1

Evaluate the limit of $k$ by plugging in $2$ for $k$ .

$\sqrt{3 \cdot 2^{2} + 4} \cdot \sqrt[3]{3 lim_{k \to 2} k + 2}$

Step 11.2

Evaluate the limit of $k$ by plugging in $2$ for $k$ .

$\sqrt{3 \cdot 2^{2} + 4} \cdot \sqrt[3]{3 \cdot 2 + 2}$

Step 12

Simplify the answer.

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Step 12.1

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

$\sqrt{3 \cdot 4 + 4} \cdot \sqrt[3]{3 \cdot 2 + 2}$

Step 12.2

Multiply $3$ by $4$ .

$\sqrt{12 + 4} \cdot \sqrt[3]{3 \cdot 2 + 2}$

Step 12.3

Add $12$ and $4$ .

$\sqrt{16} \cdot \sqrt[3]{3 \cdot 2 + 2}$

Step 12.4

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2}} \cdot \sqrt[3]{3 \cdot 2 + 2}$

Step 12.5

Pull terms out from under the radical, assuming positive real numbers.

$4 \cdot \sqrt[3]{3 \cdot 2 + 2}$

Step 12.6

Multiply $3$ by $2$ .

$4 \cdot \sqrt[3]{6 + 2}$

Step 12.7

Add $6$ and $2$ .

$4 \cdot \sqrt[3]{8}$

Step 12.8

Rewrite $8$ as $2^{3}$ .

$4 \cdot \sqrt[3]{2^{3}}$

Step 12.9

Pull terms out from under the radical, assuming real numbers.

$4 \cdot 2$

Step 12.10

Multiply $4$ by $2$ .

$8$