Calculus Examples

$\frac{lim_{n \to 8} 4^{n} - 2^{n}}{2 (4^{n}) - 2^{n}}$

Step 1

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

$\frac{lim_{n \to 8} 4^{n} - lim_{n \to 8} 2^{n}}{2 (4^{n}) - 2^{n}}$

Step 2

Move the limit into the exponent.

$\frac{4^{lim_{n \to 8} n} - lim_{n \to 8} 2^{n}}{2 (4^{n}) - 2^{n}}$

Step 3

Move the limit into the exponent.

$\frac{4^{lim_{n \to 8} n} - 2^{lim_{n \to 8} n}}{2 (4^{n}) - 2^{n}}$

Step 4

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

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Step 4.1

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

$\frac{4^{8} - 2^{lim_{n \to 8} n}}{2 (4^{n}) - 2^{n}}$

Step 4.2

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

$\frac{4^{8} - 2^{8}}{2 (4^{n}) - 2^{n}}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $4$ to the power of $8$ .

$\frac{65536 - 2^{8}}{2 \cdot 4^{n} - 2^{n}}$

Step 5.1.2

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

$\frac{65536 - 1 \cdot 256}{2 \cdot 4^{n} - 2^{n}}$

Step 5.1.3

Multiply $- 1$ by $256$ .

$\frac{65536 - 256}{2 \cdot 4^{n} - 2^{n}}$

Step 5.1.4

Subtract $256$ from $65536$ .

$\frac{65280}{2 \cdot 4^{n} - 2^{n}}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Rewrite $4^{n}$ as ${(2^{2})}^{n}$ .

$\frac{65280}{2 \cdot {(2^{2})}^{n} - 2^{n}}$

Step 5.2.2

Rewrite ${(2^{2})}^{n}$ as ${(2^{n})}^{2}$ .

$\frac{65280}{2 \cdot {(2^{n})}^{2} - 2^{n}}$

Step 5.2.3

Factor $u$ out of $2 u^{2} - u$ .

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Step 5.2.3.1

Factor $u$ out of $2 u^{2}$ .

$\frac{65280}{u (2 u) - u}$

Step 5.2.3.2

Factor $u$ out of $- u$ .

$\frac{65280}{u (2 u) + u \cdot - 1}$

Step 5.2.3.3

Factor $u$ out of $u (2 u) + u \cdot - 1$ .

$\frac{65280}{u (2 u - 1)}$

Step 5.2.4

Replace all occurrences of $u$ with $2^{n}$ .

$\frac{65280}{2^{n} (2 \cdot 2^{n} - 1)}$

Step 5.2.5

Multiply $2$ by $2^{n}$ .

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Step 5.2.5.1

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

$\frac{65280}{2^{n} (2^{1} \cdot 2^{n} - 1)}$

Step 5.2.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{65280}{2^{n} (2^{1 + n} - 1)}$