Calculus Examples

$lim_{n \to \infty} \frac{n}{\sqrt{n}}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{n \to \infty} n}{lim_{n \to \infty} \sqrt{n}}$

Step 1.1.2

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$\frac{\infty}{lim_{n \to \infty} \sqrt{n}}$

Step 1.1.3

As $n$ approaches $\infty$ for radicals, the value goes to $\infty$ .

$\frac{\infty}{\infty}$

Step 1.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{\infty}{\infty}$

Step 1.2

Since $\frac{\infty}{\infty}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{n \to \infty} \frac{n}{\sqrt{n}} = lim_{n \to \infty} \frac{\frac{d}{d n} [n]}{\frac{d}{d n} [\sqrt{n}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{n \to \infty} \frac{\frac{d}{d n} [n]}{\frac{d}{d n} [\sqrt{n}]}$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d n} [n^{n}]$ is $n \cdot n^{n - 1}$ where $n = 1$ .

$lim_{n \to \infty} \frac{1}{\frac{d}{d n} [\sqrt{n}]}$

Step 1.3.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{n}$ as $n^{\frac{1}{2}}$ .

$lim_{n \to \infty} \frac{1}{\frac{d}{d n} [n^{\frac{1}{2}}]}$

Step 1.3.4

Differentiate using the Power Rule which states that $\frac{d}{d n} [n^{n}]$ is $n \cdot n^{n - 1}$ where $n = \frac{1}{2}$ .

$lim_{n \to \infty} \frac{1}{\frac{1}{2} n^{\frac{1}{2} - 1}}$

Step 1.3.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{n \to \infty} \frac{1}{\frac{1}{2} n^{\frac{1}{2} - 1 \cdot \frac{2}{2}}}$

Step 1.3.6

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 1.3.7

Combine the numerators over the common denominator.

$lim_{n \to \infty} \frac{1}{\frac{1}{2} n^{\frac{1 - 1 \cdot 2}{2}}}$

Step 1.3.8

Simplify the numerator.

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Step 1.3.8.1

Multiply $- 1$ by $2$ .

$lim_{n \to \infty} \frac{1}{\frac{1}{2} n^{\frac{1 - 2}{2}}}$

Step 1.3.8.2

Subtract $2$ from $1$ .

$lim_{n \to \infty} \frac{1}{\frac{1}{2} n^{\frac{- 1}{2}}}$

Step 1.3.9

Move the negative in front of the fraction.

$lim_{n \to \infty} \frac{1}{\frac{1}{2} n^{- \frac{1}{2}}}$

Step 1.3.10

Simplify.

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Step 1.3.10.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.3.10.2

Multiply $\frac{1}{2}$ by $\frac{1}{n^{\frac{1}{2}}}$ .

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{n \to \infty} 1 (2 n^{\frac{1}{2}})$

Step 1.5

Rewrite $n^{\frac{1}{2}}$ as $\sqrt{n}$ .

$lim_{n \to \infty} 1 (2 \sqrt{n})$

Step 1.6

Multiply $2$ by $1$ .

$lim_{n \to \infty} 2 \sqrt{n}$

Step 2

Since the function $\sqrt{n}$ approaches $\infty$ , the positive constant $2$ times the function also approaches $\infty$ .

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Step 2.1

Consider the limit with the constant multiple $2$ removed.

$lim_{n \to \infty} \sqrt{n}$

Step 2.2

As $n$ approaches $\infty$ for radicals, the value goes to $\infty$ .

$\infty$