Calculus Examples

$lim_{k \to 8} \frac{{(k + 1)}^{k + 1}}{(k + 1) i} \cdot \frac{k i}{k^{k}}$

Step 1

Evaluate the limit.

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Step 1.1

Cancel the common factor of ${(k + 1)}^{k + 1}$ and $k + 1$ .

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Step 1.1.1

Factor $k + 1$ out of ${(k + 1)}^{k + 1}$ .

$lim_{k \to 8} \frac{(k + 1) {(k + 1)}^{k}}{(k + 1) i} \cdot \frac{k i}{k^{k}}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $k + 1$ out of $(k + 1) i$ .

$lim_{k \to 8} \frac{(k + 1) {(k + 1)}^{k}}{(k + 1) (i)} \cdot \frac{k i}{k^{k}}$

Step 1.1.2.2

Cancel the common factor.

$lim_{k \to 8} \frac{(k + 1) {(k + 1)}^{k}}{(k + 1) i} \cdot \frac{k i}{k^{k}}$

Step 1.1.2.3

Rewrite the expression.

$lim_{k \to 8} \frac{{(k + 1)}^{k}}{i} \cdot \frac{k i}{k^{k}}$

Step 1.2

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

$lim_{k \to 8} \frac{{(k + 1)}^{k}}{i} \cdot lim_{k \to 8} \frac{k i}{k^{k}}$

Step 1.3

Move the term $\frac{1}{i}$ outside of the limit because it is constant with respect to $k$ .

$\frac{1}{i} lim_{k \to 8} {(k + 1)}^{k} \cdot lim_{k \to 8} \frac{k i}{k^{k}}$

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite ${(k + 1)}^{k}$ as $e^{\ln ({(k + 1)}^{k})}$ .

$\frac{1}{i} lim_{k \to 8} e^{\ln ({(k + 1)}^{k})} \cdot lim_{k \to 8} \frac{k i}{k^{k}}$

Step 2.2

Expand $\ln ({(k + 1)}^{k})$ by moving $k$ outside the logarithm.

$\frac{1}{i} lim_{k \to 8} e^{k \ln (k + 1)} \cdot lim_{k \to 8} \frac{k i}{k^{k}}$

Step 3

Evaluate the limit.

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Step 3.1

Move the limit into the exponent.

$\frac{1}{i} e^{lim_{k \to 8} k \ln (k + 1)} \cdot lim_{k \to 8} \frac{k i}{k^{k}}$

Step 3.2

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

$\frac{1}{i} e^{lim_{k \to 8} k \cdot lim_{k \to 8} \ln (k + 1)} \cdot lim_{k \to 8} \frac{k i}{k^{k}}$

Step 3.3

Move the limit inside the logarithm.

$\frac{1}{i} e^{lim_{k \to 8} k \cdot \ln (lim_{k \to 8} k + 1)} \cdot lim_{k \to 8} \frac{k i}{k^{k}}$

Step 3.4

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

$\frac{1}{i} e^{lim_{k \to 8} k \cdot \ln (lim_{k \to 8} k + lim_{k \to 8} 1)} \cdot lim_{k \to 8} \frac{k i}{k^{k}}$

Step 3.5

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

$\frac{1}{i} e^{lim_{k \to 8} k \cdot \ln (lim_{k \to 8} k + 1)} \cdot lim_{k \to 8} \frac{k i}{k^{k}}$

Step 3.6

Move the term $i$ outside of the limit because it is constant with respect to $k$ .

$\frac{1}{i} e^{lim_{k \to 8} k \cdot \ln (lim_{k \to 8} k + 1)} \cdot i lim_{k \to 8} \frac{k}{k^{k}}$

Step 3.7

Split the limit using the Limits Quotient Rule on the limit as $k$ approaches $8$ .

$\frac{1}{i} e^{lim_{k \to 8} k \cdot \ln (lim_{k \to 8} k + 1)} \cdot i \frac{lim_{k \to 8} k}{lim_{k \to 8} k^{k}}$

Step 4

Use the properties of logarithms to simplify the limit.

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Step 4.1

Rewrite $k^{k}$ as $e^{\ln (k^{k})}$ .

$\frac{1}{i} e^{lim_{k \to 8} k \cdot \ln (lim_{k \to 8} k + 1)} \cdot i \frac{lim_{k \to 8} k}{lim_{k \to 8} e^{\ln (k^{k})}}$

Step 4.2

Expand $\ln (k^{k})$ by moving $k$ outside the logarithm.

$\frac{1}{i} e^{lim_{k \to 8} k \cdot \ln (lim_{k \to 8} k + 1)} \cdot i \frac{lim_{k \to 8} k}{lim_{k \to 8} e^{k \ln (k)}}$

Step 5

Evaluate the limit.

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Step 5.1

Move the limit into the exponent.

$\frac{1}{i} e^{lim_{k \to 8} k \cdot \ln (lim_{k \to 8} k + 1)} \cdot i \frac{lim_{k \to 8} k}{e^{lim_{k \to 8} k \ln (k)}}$

Step 5.2

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

$\frac{1}{i} e^{lim_{k \to 8} k \cdot \ln (lim_{k \to 8} k + 1)} \cdot i \frac{lim_{k \to 8} k}{e^{lim_{k \to 8} k \cdot lim_{k \to 8} \ln (k)}}$

Step 5.3

Move the limit inside the logarithm.

$\frac{1}{i} e^{lim_{k \to 8} k \cdot \ln (lim_{k \to 8} k + 1)} \cdot i \frac{lim_{k \to 8} k}{e^{lim_{k \to 8} k \cdot \ln (lim_{k \to 8} k)}}$

Step 6

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

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Step 6.1