Calculus Examples

$lim_{x \to \infty} \frac{2 \sqrt{5 x}}{5 x}$

Step 1

Move the term $\frac{2}{5}$ outside of the limit because it is constant with respect to $x$ .

$\frac{2}{5} lim_{x \to \infty} \frac{\sqrt{5 x}}{x}$

Step 2

Apply L'Hospital's rule.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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

$\frac{2}{5} \cdot \frac{lim_{x \to \infty} \sqrt{5 x}}{lim_{x \to \infty} x}$

Step 2.1.2

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

$\frac{2}{5} \cdot \frac{\infty}{lim_{x \to \infty} x}$

Step 2.1.3

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

$\frac{2}{5} \cdot \frac{\infty}{\infty}$

Step 2.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{2}{5} \cdot \frac{\infty}{\infty}$

Step 2.2

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.

$lim_{x \to \infty} \frac{\sqrt{5 x}}{x} = lim_{x \to \infty} \frac{\frac{d}{d x} [\sqrt{5 x}]}{\frac{d}{d x} [x]}$

Step 2.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 2.3.1

Differentiate the numerator and denominator.

$\frac{2}{5} lim_{x \to \infty} \frac{\frac{d}{d x} [\sqrt{5 x}]}{\frac{d}{d x} [x]}$

Step 2.3.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x}$ as ${(5 x)}^{\frac{1}{2}}$ .

$\frac{2}{5} lim_{x \to \infty} \frac{\frac{d}{d x} [{(5 x)}^{\frac{1}{2}}]}{\frac{d}{d x} [x]}$

Step 2.3.3

Factor $5$ out of $5 x$ .

$\frac{2}{5} lim_{x \to \infty} \frac{\frac{d}{d x} [{(5 (x))}^{\frac{1}{2}}]}{\frac{d}{d x} [x]}$

Step 2.3.4

Apply the product rule to $5 (x)$ .

$\frac{2}{5} lim_{x \to \infty} \frac{\frac{d}{d x} [5^{\frac{1}{2}} x^{\frac{1}{2}}]}{\frac{d}{d x} [x]}$

Step 2.3.5

Since $5^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $5^{\frac{1}{2}} x^{\frac{1}{2}}$ with respect to $x$ is $5^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$\frac{2}{5} lim_{x \to \infty} \frac{5^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]}{\frac{d}{d x} [x]}$

Step 2.3.6

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

$\frac{2}{5} lim_{x \to \infty} \frac{5^{\frac{1}{2}} \frac{1}{2} x^{\frac{1}{2} - 1}}{\frac{d}{d x} [x]}$

Step 2.3.7

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

$\frac{2}{5} lim_{x \to \infty} \frac{5^{\frac{1}{2}} \frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}}{\frac{d}{d x} [x]}$

Step 2.3.8

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

$\frac{2}{5} lim_{x \to \infty} \frac{5^{\frac{1}{2}} \frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{\frac{d}{d x} [x]}$

Step 2.3.9

Combine the numerators over the common denominator.

$\frac{2}{5} lim_{x \to \infty} \frac{5^{\frac{1}{2}} \frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}}{\frac{d}{d x} [x]}$

Step 2.3.10

Simplify the numerator.

Tap for more steps...

Step 2.3.10.1

Multiply $- 1$ by $2$ .

$\frac{2}{5} lim_{x \to \infty} \frac{5^{\frac{1}{2}} \frac{1}{2} x^{\frac{1 - 2}{2}}}{\frac{d}{d x} [x]}$

Step 2.3.10.2

Subtract $2$ from $1$ .

$\frac{2}{5} lim_{x \to \infty} \frac{5^{\frac{1}{2}} \frac{1}{2} x^{\frac{- 1}{2}}}{\frac{d}{d x} [x]}$

Step 2.3.11

Move the negative in front of the fraction.

$\frac{2}{5} lim_{x \to \infty} \frac{5^{\frac{1}{2}} \frac{1}{2} x^{- \frac{1}{2}}}{\frac{d}{d x} [x]}$

Step 2.3.12

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

$\frac{2}{5} lim_{x \to \infty} \frac{5^{\frac{1}{2}} \frac{x^{- \frac{1}{2}}}{2}}{\frac{d}{d x} [x]}$

Step 2.3.13

Combine $5^{\frac{1}{2}}$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{2}{5} lim_{x \to \infty} \frac{\frac{5^{\frac{1}{2}} x^{- \frac{1}{2}}}{2}}{\frac{d}{d x} [x]}$

Step 2.3.14

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{5} lim_{x \to \infty} \frac{\frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [x]}$

Step 2.3.15

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

$\frac{2}{5} lim_{x \to \infty} \frac{\frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}}{1}$

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{2}{5} lim_{x \to \infty} \frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} \cdot 1$

Step 2.5

Convert fractional exponents to radicals.

Tap for more steps...

Step 2.5.1

Rewrite $5^{\frac{1}{2}}$ as $\sqrt{5}$ .

$\frac{2}{5} lim_{x \to \infty} \frac{\sqrt{5}}{2 x^{\frac{1}{2}}} \cdot 1$

Step 2.5.2

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

$\frac{2}{5} lim_{x \to \infty} \frac{\sqrt{5}}{2 \sqrt{x}} \cdot 1$

Step 2.6

Multiply $\frac{\sqrt{5}}{2 \sqrt{x}}$ by $1$ .

$\frac{2}{5} lim_{x \to \infty} \frac{\sqrt{5}}{2 \sqrt{x}}$

Step 3

Move the term $\frac{\sqrt{5}}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{2}{5} \cdot \frac{\sqrt{5}}{2} lim_{x \to \infty} \frac{1}{\sqrt{x}}$

Step 4

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

$\frac{2}{5} \cdot \frac{\sqrt{5}}{2} \cdot 0$

Step 5

Simplify the answer.

Tap for more steps...

Step 5.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.1.1

Cancel the common factor.

$\frac{2}{5} \cdot \frac{\sqrt{5}}{2} \cdot 0$

Step 5.1.2

Rewrite the expression.

$\frac{1}{5} \sqrt{5} \cdot 0$

Step 5.2

Combine $\frac{1}{5}$ and $\sqrt{5}$ .

$\frac{\sqrt{5}}{5} \cdot 0$

Step 5.3

Multiply $\frac{\sqrt{5}}{5}$ by $0$ .

$0$