Calculus Examples

$lim_{x \to - 6} {(2 t + 11)}^{5}$

Step 1

Evaluate the limit of ${(2 t + 11)}^{5}$ which is constant as $x$ approaches $- 6$ .

${(2 t + 11)}^{5}$

Step 2

Simplify the answer.

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Step 2.1

Use the Binomial Theorem.

${(2 t)}^{5} + 5 {(2 t)}^{4} \cdot 11 + 10 {(2 t)}^{3} \cdot 11^{2} + 10 {(2 t)}^{2} \cdot 11^{3} + 5 (2 t) \cdot 11^{4} + 11^{5}$

Step 2.2

Simplify each term.

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Step 2.2.1

Apply the product rule to $2 t$ .

$2^{5} t^{5} + 5 {(2 t)}^{4} \cdot 11 + 10 {(2 t)}^{3} \cdot 11^{2} + 10 {(2 t)}^{2} \cdot 11^{3} + 5 (2 t) \cdot 11^{4} + 11^{5}$

Step 2.2.2

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

$32 t^{5} + 5 {(2 t)}^{4} \cdot 11 + 10 {(2 t)}^{3} \cdot 11^{2} + 10 {(2 t)}^{2} \cdot 11^{3} + 5 (2 t) \cdot 11^{4} + 11^{5}$

Step 2.2.3

Apply the product rule to $2 t$ .

$32 t^{5} + 5 (2^{4} t^{4}) \cdot 11 + 10 {(2 t)}^{3} \cdot 11^{2} + 10 {(2 t)}^{2} \cdot 11^{3} + 5 (2 t) \cdot 11^{4} + 11^{5}$

Step 2.2.4

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

$32 t^{5} + 5 (16 t^{4}) \cdot 11 + 10 {(2 t)}^{3} \cdot 11^{2} + 10 {(2 t)}^{2} \cdot 11^{3} + 5 (2 t) \cdot 11^{4} + 11^{5}$

Step 2.2.5

Multiply $16$ by $5$ .

$32 t^{5} + 80 t^{4} \cdot 11 + 10 {(2 t)}^{3} \cdot 11^{2} + 10 {(2 t)}^{2} \cdot 11^{3} + 5 (2 t) \cdot 11^{4} + 11^{5}$

Step 2.2.6

Multiply $11$ by $80$ .

$32 t^{5} + 880 t^{4} + 10 {(2 t)}^{3} \cdot 11^{2} + 10 {(2 t)}^{2} \cdot 11^{3} + 5 (2 t) \cdot 11^{4} + 11^{5}$

Step 2.2.7

Apply the product rule to $2 t$ .

$32 t^{5} + 880 t^{4} + 10 (2^{3} t^{3}) \cdot 11^{2} + 10 {(2 t)}^{2} \cdot 11^{3} + 5 (2 t) \cdot 11^{4} + 11^{5}$

Step 2.2.8

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

$32 t^{5} + 880 t^{4} + 10 (8 t^{3}) \cdot 11^{2} + 10 {(2 t)}^{2} \cdot 11^{3} + 5 (2 t) \cdot 11^{4} + 11^{5}$

Step 2.2.9

Multiply $8$ by $10$ .

$32 t^{5} + 880 t^{4} + 80 t^{3} \cdot 11^{2} + 10 {(2 t)}^{2} \cdot 11^{3} + 5 (2 t) \cdot 11^{4} + 11^{5}$

Step 2.2.10

Raise $11$ to the power of $2$ .

$32 t^{5} + 880 t^{4} + 80 t^{3} \cdot 121 + 10 {(2 t)}^{2} \cdot 11^{3} + 5 (2 t) \cdot 11^{4} + 11^{5}$

Step 2.2.11

Multiply $121$ by $80$ .

$32 t^{5} + 880 t^{4} + 9680 t^{3} + 10 {(2 t)}^{2} \cdot 11^{3} + 5 (2 t) \cdot 11^{4} + 11^{5}$

Step 2.2.12

Apply the product rule to $2 t$ .

$32 t^{5} + 880 t^{4} + 9680 t^{3} + 10 (2^{2} t^{2}) \cdot 11^{3} + 5 (2 t) \cdot 11^{4} + 11^{5}$

Step 2.2.13

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

$32 t^{5} + 880 t^{4} + 9680 t^{3} + 10 (4 t^{2}) \cdot 11^{3} + 5 (2 t) \cdot 11^{4} + 11^{5}$

Step 2.2.14

Multiply $4$ by $10$ .

$32 t^{5} + 880 t^{4} + 9680 t^{3} + 40 t^{2} \cdot 11^{3} + 5 (2 t) \cdot 11^{4} + 11^{5}$

Step 2.2.15

Raise $11$ to the power of $3$ .

$32 t^{5} + 880 t^{4} + 9680 t^{3} + 40 t^{2} \cdot 1331 + 5 (2 t) \cdot 11^{4} + 11^{5}$

Step 2.2.16

Multiply $1331$ by $40$ .

$32 t^{5} + 880 t^{4} + 9680 t^{3} + 53240 t^{2} + 5 (2 t) \cdot 11^{4} + 11^{5}$

Step 2.2.17

Multiply $2$ by $5$ .

$32 t^{5} + 880 t^{4} + 9680 t^{3} + 53240 t^{2} + 10 t \cdot 11^{4} + 11^{5}$

Step 2.2.18

Raise $11$ to the power of $4$ .

$32 t^{5} + 880 t^{4} + 9680 t^{3} + 53240 t^{2} + 10 t \cdot 14641 + 11^{5}$

Step 2.2.19

Multiply $14641$ by $10$ .

$32 t^{5} + 880 t^{4} + 9680 t^{3} + 53240 t^{2} + 146410 t + 11^{5}$

Step 2.2.20

Raise $11$ to the power of $5$ .

$32 t^{5} + 880 t^{4} + 9680 t^{3} + 53240 t^{2} + 146410 t + 161051$