Calculus Examples

$lim_{t \to - 1} {(t^{2} + 1)}^{4} {(t + 3)}^{5}$

Step 1

Split the limit using the Product of Limits Rule on the limit as $t$ approaches $- 1$ .

$lim_{t \to - 1} {(t^{2} + 1)}^{4} \cdot lim_{t \to - 1} {(t + 3)}^{5}$

Step 2

Move the exponent $4$ from ${(t^{2} + 1)}^{4}$ outside the limit using the Limits Power Rule.

${(lim_{t \to - 1} t^{2} + 1)}^{4} \cdot lim_{t \to - 1} {(t + 3)}^{5}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $- 1$ .

${(lim_{t \to - 1} t^{2} + lim_{t \to - 1} 1)}^{4} \cdot lim_{t \to - 1} {(t + 3)}^{5}$

Step 4

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

${({(lim_{t \to - 1} t)}^{2} + lim_{t \to - 1} 1)}^{4} \cdot lim_{t \to - 1} {(t + 3)}^{5}$

Step 5

Evaluate the limit of $1$ which is constant as $t$ approaches $- 1$ .

${({(lim_{t \to - 1} t)}^{2} + 1)}^{4} \cdot lim_{t \to - 1} {(t + 3)}^{5}$

Step 6

Move the exponent $5$ from ${(t + 3)}^{5}$ outside the limit using the Limits Power Rule.

${({(lim_{t \to - 1} t)}^{2} + 1)}^{4} \cdot {(lim_{t \to - 1} t + 3)}^{5}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $- 1$ .

${({(lim_{t \to - 1} t)}^{2} + 1)}^{4} \cdot {(lim_{t \to - 1} t + lim_{t \to - 1} 3)}^{5}$

Step 8

Evaluate the limit of $3$ which is constant as $t$ approaches $- 1$ .

${({(lim_{t \to - 1} t)}^{2} + 1)}^{4} \cdot {(lim_{t \to - 1} t + 3)}^{5}$

Step 9

Evaluate the limits by plugging in $- 1$ for all occurrences of $t$ .

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Step 9.1

Evaluate the limit of $t$ by plugging in $- 1$ for $t$ .

${({(- 1)}^{2} + 1)}^{4} \cdot {(lim_{t \to - 1} t + 3)}^{5}$

Step 9.2

Evaluate the limit of $t$ by plugging in $- 1$ for $t$ .

${({(- 1)}^{2} + 1)}^{4} \cdot {(- 1 + 3)}^{5}$

Step 10

Simplify the answer.

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Step 10.1

Raise $- 1$ to the power of $2$ .

${(1 + 1)}^{4} \cdot {(- 1 + 3)}^{5}$

Step 10.2

Add $1$ and $1$ .

$2^{4} \cdot {(- 1 + 3)}^{5}$

Step 10.3

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

$16 \cdot {(- 1 + 3)}^{5}$

Step 10.4

Add $- 1$ and $3$ .

$16 \cdot 2^{5}$

Step 10.5

Raise $2$ to the power of $5$ .

$16 \cdot 32$

Step 10.6

Multiply $16$ by $32$ .

$512$