Calculus Examples

$lim_{n \to 3} \frac{8^{\log_{2} (n)}}{2^{n}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $n$ approaches $3$ .

$\frac{lim_{n \to 3} 8^{\log_{2} (n)}}{lim_{n \to 3} 2^{n}}$

Step 2

Move the limit into the exponent.

$\frac{8^{lim_{n \to 3} \log_{2} (n)}}{lim_{n \to 3} 2^{n}}$

Step 3

Move the limit inside the logarithm.

$\frac{8^{\log_{2} (lim_{n \to 3} n)}}{lim_{n \to 3} 2^{n}}$

Step 4

Move the limit into the exponent.

$\frac{8^{\log_{2} (lim_{n \to 3} n)}}{2^{lim_{n \to 3} n}}$

Step 5

Evaluate the limits by plugging in $3$ for all occurrences of $n$ .

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Step 5.1

Evaluate the limit of $n$ by plugging in $3$ for $n$ .

$\frac{8^{\log_{2} (3)}}{2^{lim_{n \to 3} n}}$

Step 5.2

Evaluate the limit of $n$ by plugging in $3$ for $n$ .

$\frac{8^{\log_{2} (3)}}{2^{3}}$

Step 6

Simplify the answer.

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Step 6.1

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

$\frac{8^{\log_{2} (3)}}{8}$

Step 6.2

Cancel the common factor of $8^{\log_{2} (3)}$ and $8$ .

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Step 6.2.1

Factor $8$ out of $8^{\log_{2} (3)}$ .

$\frac{8 \cdot 8^{\log_{2} (3) - 1}}{8}$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $8$ out of $8$ .

$\frac{8 \cdot 8^{\log_{2} (3) - 1}}{8 (1)}$

Step 6.2.2.2

Cancel the common factor.

$\frac{8 \cdot 8^{\log_{2} (3) - 1}}{8 \cdot 1}$

Step 6.2.2.3

Rewrite the expression.

$\frac{8^{\log_{2} (3) - 1}}{1}$

Step 6.2.2.4

Divide $8^{\log_{2} (3) - 1}$ by $1$ .

$8^{\log_{2} (3) - 1}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$8^{\log_{2} (3) - 1}$

Decimal Form:

$3.375$