Calculus Examples

$lim_{n \to 8} (3^{\sin (n \cdot π)} - 1) \cdot \ln (n^{2} + 2)$

Step 1

Split the limit using the Product of Limits Rule on the limit as $n$ approaches $8$ .

$lim_{n \to 8} 3^{\sin (n \cdot π)} - 1 \cdot lim_{n \to 8} \ln (n^{2} + 2)$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $n$ approaches $8$ .

$(lim_{n \to 8} 3^{\sin (n \cdot π)} - lim_{n \to 8} 1) \cdot lim_{n \to 8} \ln (n^{2} + 2)$

Step 3

Move the limit into the exponent.

$(3^{lim_{n \to 8} \sin (n \cdot π)} - lim_{n \to 8} 1) \cdot lim_{n \to 8} \ln (n^{2} + 2)$

Step 4

Move the limit inside the trig function because sine is continuous.

$(3^{\sin (lim_{n \to 8} n \cdot π)} - lim_{n \to 8} 1) \cdot lim_{n \to 8} \ln (n^{2} + 2)$

Step 5

Move the term $π$ outside of the limit because it is constant with respect to $n$ .

$(3^{\sin (π lim_{n \to 8} n)} - lim_{n \to 8} 1) \cdot lim_{n \to 8} \ln (n^{2} + 2)$

Step 6

Evaluate the limit of $1$ which is constant as $n$ approaches $8$ .

$(3^{\sin (π lim_{n \to 8} n)} - 1 \cdot 1) \cdot lim_{n \to 8} \ln (n^{2} + 2)$

Step 7

Move the limit inside the logarithm.

$(3^{\sin (π lim_{n \to 8} n)} - 1 \cdot 1) \cdot \ln (lim_{n \to 8} n^{2} + 2)$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $n$ approaches $8$ .

$(3^{\sin (π lim_{n \to 8} n)} - 1 \cdot 1) \cdot \ln (lim_{n \to 8} n^{2} + lim_{n \to 8} 2)$

Step 9

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

$(3^{\sin (π lim_{n \to 8} n)} - 1 \cdot 1) \cdot \ln ({(lim_{n \to 8} n)}^{2} + lim_{n \to 8} 2)$

Step 10

Evaluate the limit of $2$ which is constant as $n$ approaches $8$ .

$(3^{\sin (π lim_{n \to 8} n)} - 1 \cdot 1) \cdot \ln ({(lim_{n \to 8} n)}^{2} + 2)$

Step 11

Evaluate the limits by plugging in $8$ for all occurrences of $n$ .

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Step 11.1

Evaluate the limit of $n$ by plugging in $8$ for $n$ .

$(3^{\sin (π \cdot 8)} - 1 \cdot 1) \cdot \ln ({(lim_{n \to 8} n)}^{2} + 2)$

Step 11.2

Evaluate the limit of $n$ by plugging in $8$ for $n$ .

$(3^{\sin (π \cdot 8)} - 1 \cdot 1) \cdot \ln (8^{2} + 2)$

Step 12

Simplify the answer.

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Step 12.1

Simplify each term.

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Step 12.1.1

Move $8$ to the left of $π$ .

$(3^{\sin (8 \cdot π)} - 1 \cdot 1) \cdot \ln (8^{2} + 2)$

Step 12.1.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$(3^{\sin (0)} - 1 \cdot 1) \cdot \ln (8^{2} + 2)$

Step 12.1.3

The exact value of $\sin (0)$ is $0$ .

$(3^{0} - 1 \cdot 1) \cdot \ln (8^{2} + 2)$

Step 12.1.4

Anything raised to $0$ is $1$ .

$(1 - 1 \cdot 1) \cdot \ln (8^{2} + 2)$

Step 12.1.5

Multiply $- 1$ by $1$ .

$(1 - 1) \cdot \ln (8^{2} + 2)$

Step 12.2

Subtract $1$ from $1$ .

$0 \cdot \ln (8^{2} + 2)$

Step 12.3

Raise $8$ to the power of $2$ .

$0 \cdot \ln (64 + 2)$

Step 12.4

Add $64$ and $2$ .

$0 \cdot \ln (66)$

Step 12.5

Multiply $0$ by $\ln (66)$ .

$0$