Calculus Examples

$lim_{t \to 3} \sqrt[3]{t^{2} + t - 4}$

Step 1

Move the limit under the radical sign.

$\sqrt[3]{lim_{t \to 3} t^{2} + t - 4}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $3$ .

$\sqrt[3]{lim_{t \to 3} t^{2} + lim_{t \to 3} t - lim_{t \to 3} 4}$

Step 3

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

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

Step 4

Evaluate the limit of $4$ which is constant as $t$ approaches $3$ .

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

Step 5

Evaluate the limits by plugging in $3$ for all occurrences of $t$ .

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Step 5.1

Evaluate the limit of $t$ by plugging in $3$ for $t$ .

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

Step 5.2

Evaluate the limit of $t$ by plugging in $3$ for $t$ .

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

Step 6

Simplify the answer.

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Step 6.1

Raise $3$ to the power of $2$ .

$\sqrt[3]{9 + 3 - 1 \cdot 4}$

Step 6.2

Multiply $- 1$ by $4$ .

$\sqrt[3]{9 + 3 - 4}$

Step 6.3

Add $9$ and $3$ .

$\sqrt[3]{12 - 4}$

Step 6.4

Subtract $4$ from $12$ .

$\sqrt[3]{8}$

Step 6.5

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

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

Step 6.6

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

$2$