Calculus Examples

$lim_{x \to 0} \frac{x}{\sqrt{1 + 3 x} - 1}$

Step 1

Apply L'Hospital's rule.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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

$\frac{lim_{x \to 0} x}{lim_{x \to 0} \sqrt{1 + 3 x} - 1}$

Step 1.1.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{lim_{x \to 0} \sqrt{1 + 3 x} - 1}$

Step 1.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 1.1.3.1

Evaluate the limit.

Tap for more steps...

Step 1.1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{0}{lim_{x \to 0} \sqrt{1 + 3 x} - lim_{x \to 0} 1}$

Step 1.1.3.1.2

Move the limit under the radical sign.

$\frac{0}{\sqrt{lim_{x \to 0} 1 + 3 x} - lim_{x \to 0} 1}$

Step 1.1.3.1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{0}{\sqrt{lim_{x \to 0} 1 + lim_{x \to 0} 3 x} - lim_{x \to 0} 1}$

Step 1.1.3.1.4

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{0}{\sqrt{1 + lim_{x \to 0} 3 x} - lim_{x \to 0} 1}$

Step 1.1.3.1.5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{\sqrt{1 + 3 lim_{x \to 0} x} - lim_{x \to 0} 1}$

Step 1.1.3.1.6

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{0}{\sqrt{1 + 3 lim_{x \to 0} x} - 1 \cdot 1}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{\sqrt{1 + 3 \cdot 0} - 1 \cdot 1}$

Step 1.1.3.3

Simplify the answer.

Tap for more steps...

Step 1.1.3.3.1

Simplify each term.

Tap for more steps...

Step 1.1.3.3.1.1

Multiply $3$ by $0$ .

$\frac{0}{\sqrt{1 + 0} - 1 \cdot 1}$

Step 1.1.3.3.1.2

Add $1$ and $0$ .

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

Step 1.1.3.3.1.3

Any root of $1$ is $1$ .

$\frac{0}{1 - 1 \cdot 1}$

Step 1.1.3.3.1.4

Multiply $- 1$ by $1$ .

$\frac{0}{1 - 1}$

Step 1.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.1.3.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ 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.

$lim_{x \to 0} \frac{x}{\sqrt{1 + 3 x} - 1} = lim_{x \to 0} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [\sqrt{1 + 3 x} - 1]}$

Step 1.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [\sqrt{1 + 3 x} - 1]}$

Step 1.3.2

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

$lim_{x \to 0} \frac{1}{\frac{d}{d x} [\sqrt{1 + 3 x} - 1]}$

Step 1.3.3

By the Sum Rule, the derivative of $\sqrt{1 + 3 x} - 1$ with respect to $x$ is $\frac{d}{d x} [\sqrt{1 + 3 x}] + \frac{d}{d x} [- 1]$ .

$lim_{x \to 0} \frac{1}{\frac{d}{d x} [\sqrt{1 + 3 x}] + \frac{d}{d x} [- 1]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [\sqrt{1 + 3 x}]$ .

Tap for more steps...

Step 1.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + 3 x}$ as ${(1 + 3 x)}^{\frac{1}{2}}$ .

$lim_{x \to 0} \frac{1}{\frac{d}{d x} [{(1 + 3 x)}^{\frac{1}{2}}] + \frac{d}{d x} [- 1]}$

Step 1.3.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 1 + 3 x$ .

Tap for more steps...

Step 1.3.4.2.1

To apply the Chain Rule, set $u$ as $1 + 3 x$ .

$lim_{x \to 0} \frac{1}{\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [1 + 3 x] + \frac{d}{d x} [- 1]}$

Step 1.3.4.2.2

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

$lim_{x \to 0} \frac{1}{\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [1 + 3 x] + \frac{d}{d x} [- 1]}$

Step 1.3.4.2.3

Replace all occurrences of $u$ with $1 + 3 x$ .

$lim_{x \to 0} \frac{1}{\frac{1}{2} {(1 + 3 x)}^{\frac{1}{2} - 1} \frac{d}{d x} [1 + 3 x] + \frac{d}{d x} [- 1]}$

Step 1.3.4.3

By the Sum Rule, the derivative of $1 + 3 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [3 x]$ .

$lim_{x \to 0} \frac{1}{\frac{1}{2} {(1 + 3 x)}^{\frac{1}{2} - 1} (\frac{d}{d x} [1] + \frac{d}{d x} [3 x]) + \frac{d}{d x} [- 1]}$

Step 1.3.4.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{1}{\frac{1}{2} {(1 + 3 x)}^{\frac{1}{2} - 1} (0 + \frac{d}{d x} [3 x]) + \frac{d}{d x} [- 1]}$

Step 1.3.4.5

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{1}{\frac{1}{2} {(1 + 3 x)}^{\frac{1}{2} - 1} (0 + 3 \frac{d}{d x} [x]) + \frac{d}{d x} [- 1]}$

Step 1.3.4.6

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

$lim_{x \to 0} \frac{1}{\frac{1}{2} {(1 + 3 x)}^{\frac{1}{2} - 1} (0 + 3 \cdot 1) + \frac{d}{d x} [- 1]}$

Step 1.3.4.7

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

$lim_{x \to 0} \frac{1}{\frac{1}{2} {(1 + 3 x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (0 + 3 \cdot 1) + \frac{d}{d x} [- 1]}$

Step 1.3.4.8

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

$lim_{x \to 0} \frac{1}{\frac{1}{2} {(1 + 3 x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (0 + 3 \cdot 1) + \frac{d}{d x} [- 1]}$

Step 1.3.4.9

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{1}{\frac{1}{2} {(1 + 3 x)}^{\frac{1 - 1 \cdot 2}{2}} (0 + 3 \cdot 1) + \frac{d}{d x} [- 1]}$

Step 1.3.4.10

Simplify the numerator.

Tap for more steps...

Step 1.3.4.10.1

Multiply $- 1$ by $2$ .

$lim_{x \to 0} \frac{1}{\frac{1}{2} {(1 + 3 x)}^{\frac{1 - 2}{2}} (0 + 3 \cdot 1) + \frac{d}{d x} [- 1]}$

Step 1.3.4.10.2

Subtract $2$ from $1$ .

$lim_{x \to 0} \frac{1}{\frac{1}{2} {(1 + 3 x)}^{\frac{- 1}{2}} (0 + 3 \cdot 1) + \frac{d}{d x} [- 1]}$

Step 1.3.4.11

Move the negative in front of the fraction.

$lim_{x \to 0} \frac{1}{\frac{1}{2} {(1 + 3 x)}^{- \frac{1}{2}} (0 + 3 \cdot 1) + \frac{d}{d x} [- 1]}$

Step 1.3.4.12

Multiply $3$ by $1$ .

$lim_{x \to 0} \frac{1}{\frac{1}{2} {(1 + 3 x)}^{- \frac{1}{2}} (0 + 3) + \frac{d}{d x} [- 1]}$

Step 1.3.4.13

Add $0$ and $3$ .

$lim_{x \to 0} \frac{1}{\frac{1}{2} {(1 + 3 x)}^{- \frac{1}{2}} \cdot 3 + \frac{d}{d x} [- 1]}$

Step 1.3.4.14

Combine $\frac{1}{2}$ and ${(1 + 3 x)}^{- \frac{1}{2}}$ .

$lim_{x \to 0} \frac{1}{\frac{{(1 + 3 x)}^{- \frac{1}{2}}}{2} \cdot 3 + \frac{d}{d x} [- 1]}$

Step 1.3.4.15

Combine $\frac{{(1 + 3 x)}^{- \frac{1}{2}}}{2}$ and $3$ .

$lim_{x \to 0} \frac{1}{\frac{{(1 + 3 x)}^{- \frac{1}{2}} \cdot 3}{2} + \frac{d}{d x} [- 1]}$

Step 1.3.4.16

Move $3$ to the left of ${(1 + 3 x)}^{- \frac{1}{2}}$ .

$lim_{x \to 0} \frac{1}{\frac{3 \cdot {(1 + 3 x)}^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 1]}$

Step 1.3.4.17

Move ${(1 + 3 x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 0} \frac{1}{\frac{3}{2 {(1 + 3 x)}^{\frac{1}{2}}} + \frac{d}{d x} [- 1]}$

Step 1.3.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{1}{\frac{3}{2 {(1 + 3 x)}^{\frac{1}{2}}} + 0}$

Step 1.3.6

Add $\frac{3}{2 {(1 + 3 x)}^{\frac{1}{2}}}$ and $0$ .

$lim_{x \to 0} \frac{1}{\frac{3}{2 {(1 + 3 x)}^{\frac{1}{2}}}}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} 1 \frac{2 {(1 + 3 x)}^{\frac{1}{2}}}{3}$

Step 1.5

Rewrite ${(1 + 3 x)}^{\frac{1}{2}}$ as $\sqrt{1 + 3 x}$ .

$lim_{x \to 0} 1 \frac{2 \sqrt{1 + 3 x}}{3}$

Step 1.6

Multiply $\frac{2 \sqrt{1 + 3 x}}{3}$ by $1$ .

$lim_{x \to 0} \frac{2 \sqrt{1 + 3 x}}{3}$

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

Move the term $\frac{2}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{2}{3} lim_{x \to 0} \sqrt{1 + 3 x}$

Step 2.2

Move the limit under the radical sign.

$\frac{2}{3} \sqrt{lim_{x \to 0} 1 + 3 x}$

Step 2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{2}{3} \sqrt{lim_{x \to 0} 1 + lim_{x \to 0} 3 x}$

Step 2.4

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{2}{3} \sqrt{1 + lim_{x \to 0} 3 x}$

Step 2.5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{2}{3} \sqrt{1 + 3 lim_{x \to 0} x}$

Step 3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{2}{3} \sqrt{1 + 3 \cdot 0}$

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

Multiply $3$ by $0$ .

$\frac{2}{3} \sqrt{1 + 0}$

Step 4.2

Add $1$ and $0$ .

$\frac{2}{3} \sqrt{1}$

Step 4.3

Any root of $1$ is $1$ .

$\frac{2}{3} \cdot 1$

Step 4.4

Multiply $\frac{2}{3}$ by $1$ .

$\frac{2}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$