Calculus Examples

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

Step 1

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

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

Step 2

Divide the numerator and denominator by the highest power of $n$ in the denominator, which is $n^{2}$ .

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

Step 3

Evaluate the limit.

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Step 3.1

Cancel the common factor of $n^{2}$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

$4 lim_{n \to \infty} \frac{1}{\frac{n^{2}}{n^{2}} + \frac{10000 n}{n^{2}}}$

Step 3.2

Simplify each term.

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Step 3.2.1

Cancel the common factor of $n^{2}$ .

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Step 3.2.1.1

Cancel the common factor.

$4 lim_{n \to \infty} \frac{1}{\frac{n^{2}}{n^{2}} + \frac{10000 n}{n^{2}}}$

Step 3.2.1.2

Rewrite the expression.

$4 lim_{n \to \infty} \frac{1}{1 + \frac{10000 n}{n^{2}}}$

Step 3.2.2

Cancel the common factor of $n$ and $n^{2}$ .

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Step 3.2.2.1

Factor $n$ out of $10000 n$ .

$4 lim_{n \to \infty} \frac{1}{1 + \frac{n \cdot 10000}{n^{2}}}$

Step 3.2.2.2

Cancel the common factors.

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Step 3.2.2.2.1

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

$4 lim_{n \to \infty} \frac{1}{1 + \frac{n \cdot 10000}{n \cdot n}}$

Step 3.2.2.2.2

Cancel the common factor.

$4 lim_{n \to \infty} \frac{1}{1 + \frac{n \cdot 10000}{n \cdot n}}$

Step 3.2.2.2.3

Rewrite the expression.

$4 lim_{n \to \infty} \frac{1}{1 + \frac{10000}{n}}$

Step 3.3

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

$4 \frac{lim_{n \to \infty} 1}{lim_{n \to \infty} 1 + \frac{10000}{n}}$

Step 3.4

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

$4 \frac{1}{lim_{n \to \infty} 1 + \frac{10000}{n}}$

Step 3.5

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

$4 \frac{1}{lim_{n \to \infty} 1 + lim_{n \to \infty} \frac{10000}{n}}$

Step 3.6

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

$4 \frac{1}{1 + lim_{n \to \infty} \frac{10000}{n}}$

Step 3.7

Move the term $10000$ outside of the limit because it is constant with respect to $n$ .

$4 \frac{1}{1 + 10000 lim_{n \to \infty} \frac{1}{n}}$

Step 4

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{n}$ approaches $0$ .

$4 \frac{1}{1 + 10000 \cdot 0}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the denominator.

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Step 5.1.1

Multiply $10000$ by $0$ .

$4 \frac{1}{1 + 0}$

Step 5.1.2

Add $1$ and $0$ .

$4 (\frac{1}{1})$

Step 5.2

Cancel the common factor of $1$ .

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Step 5.2.1

Cancel the common factor.

$4 (\frac{1}{1})$

Step 5.2.2

Rewrite the expression.

$4 \cdot 1$

Step 5.3

Multiply $4$ by $1$ .

$4$