Calculus Examples

$\frac{lim_{n \to 8} 4 n^{2}}{n^{2} + 10000 n}$

Step 1

Evaluate the limit.

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Step 1.1

Move the term $4$ outside of the limit because it is constant with respect to $n$ .

$\frac{4 lim_{n \to 8} n^{2}}{n^{2} + 10000 n}$

Step 1.2

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

$\frac{4 {(lim_{n \to 8} n)}^{2}}{n^{2} + 10000 n}$

Step 2

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

$\frac{4 \cdot 8^{2}}{n^{2} + 10000 n}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Rewrite $4$ as $2^{2}$ .

$\frac{2^{2} \cdot 8^{2}}{n^{2} + 10000 n}$

Step 3.1.2

Rewrite $8$ as $2^{3}$ .

$\frac{2^{2} \cdot {(2^{3})}^{2}}{n^{2} + 10000 n}$

Step 3.1.3

Multiply the exponents in ${(2^{3})}^{2}$ .

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Step 3.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2^{2} \cdot 2^{3 \cdot 2}}{n^{2} + 10000 n}$

Step 3.1.3.2

Multiply $3$ by $2$ .

$\frac{2^{2} \cdot 2^{6}}{n^{2} + 10000 n}$

Step 3.1.4

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

$\frac{2^{2 + 6}}{n^{2} + 10000 n}$

Step 3.1.5

Add $2$ and $6$ .

$\frac{2^{8}}{n^{2} + 10000 n}$

Step 3.2

Factor $n$ out of $n^{2} + 10000 n$ .

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Step 3.2.1

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

$\frac{2^{8}}{n \cdot n + 10000 n}$

Step 3.2.2

Factor $n$ out of $10000 n$ .

$\frac{2^{8}}{n \cdot n + n \cdot 10000}$

Step 3.2.3

Factor $n$ out of $n \cdot n + n \cdot 10000$ .

$\frac{2^{8}}{n (n + 10000)}$

Step 3.3

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

$\frac{256}{n (n + 10000)}$