Calculus Examples

$lim_{t \to - 2} t^{3} (6 - 4 t^{2})$

Step 1

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

$lim_{t \to - 2} t^{3} \cdot lim_{t \to - 2} 6 - 4 t^{2}$

Step 2

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

${(lim_{t \to - 2} t)}^{3} \cdot lim_{t \to - 2} 6 - 4 t^{2}$

Step 3

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

${(lim_{t \to - 2} t)}^{3} \cdot (lim_{t \to - 2} 6 - lim_{t \to - 2} 4 t^{2})$

Step 4

Evaluate the limit of $6$ which is constant as $t$ approaches $- 2$ .

${(lim_{t \to - 2} t)}^{3} \cdot (6 - lim_{t \to - 2} 4 t^{2})$

Step 5

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

${(lim_{t \to - 2} t)}^{3} \cdot (6 - 4 lim_{t \to - 2} t^{2})$

Step 6

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

${(lim_{t \to - 2} t)}^{3} \cdot (6 - 4 {(lim_{t \to - 2} t)}^{2})$

Step 7

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

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Step 7.1

Evaluate the limit of $t$ by plugging in $- 2$ for $t$ .

${(- 2)}^{3} \cdot (6 - 4 {(lim_{t \to - 2} t)}^{2})$

Step 7.2

Evaluate the limit of $t$ by plugging in $- 2$ for $t$ .

${(- 2)}^{3} \cdot (6 - 4 {(- 2)}^{2})$

Step 8

Simplify the answer.

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Step 8.1

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

$- 8 \cdot (6 - 4 {(- 2)}^{2})$

Step 8.2

Simplify each term.

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Step 8.2.1

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

$- 8 \cdot (6 - 4 \cdot 4)$

Step 8.2.2

Multiply $- 4$ by $4$ .

$- 8 \cdot (6 - 16)$

Step 8.3

Subtract $16$ from $6$ .

$- 8 \cdot - 10$

Step 8.4

Multiply $- 8$ by $- 10$ .

$80$