Calculus Examples

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

Step 1

Combine terms.

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Step 1.1

To write $\frac{1}{x \sqrt{1} + x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 1.2

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

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

Step 1.3

Write each expression with a common denominator of $(x \sqrt{1} + x) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Reorder the factors of $(x \sqrt{1} + x) x$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 2.1.2.1

Evaluate the limit of $x - (x \sqrt{1} + x)$ by plugging in $0$ for $x$ .

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

Step 2.1.2.2

Simplify each term.

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Step 2.1.2.2.1

Simplify each term.

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Step 2.1.2.2.1.1

Any root of $1$ is $1$ .

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

Step 2.1.2.2.1.2

Multiply $0$ by $1$ .

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

Step 2.1.2.2.2

Add $0$ and $0$ .

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

Step 2.1.2.2.3

Multiply $- 1$ by $0$ .

$\frac{0 + 0}{lim_{x \to 0} x (x \sqrt{1} + x)}$

Step 2.1.2.3

Add $0$ and $0$ .

$\frac{0}{lim_{x \to 0} x (x \sqrt{1} + x)}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

$\frac{0}{lim_{x \to 0} x \cdot (lim_{x \to 0} x \sqrt{1} + lim_{x \to 0} x)}$

Step 2.1.3.3

Move the term $\sqrt{1}$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{lim_{x \to 0} x \cdot (\sqrt{1} lim_{x \to 0} x + lim_{x \to 0} x)}$

Step 2.1.3.4

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 2.1.3.4.1

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

$\frac{0}{0 \cdot (\sqrt{1} lim_{x \to 0} x + lim_{x \to 0} x)}$

Step 2.1.3.4.2

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

$\frac{0}{0 \cdot (\sqrt{1} \cdot 0 + lim_{x \to 0} x)}$

Step 2.1.3.4.3

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

$\frac{0}{0 \cdot (\sqrt{1} \cdot 0 + 0)}$

Step 2.1.3.5

Simplify the answer.

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Step 2.1.3.5.1

Simplify each term.

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Step 2.1.3.5.1.1

Any root of $1$ is $1$ .

$\frac{0}{0 \cdot (1 \cdot 0 + 0)}$

Step 2.1.3.5.1.2

Multiply $0$ by $1$ .

$\frac{0}{0 \cdot (0 + 0)}$

Step 2.1.3.5.2

Add $0$ and $0$ .

$\frac{0}{0 \cdot 0}$

Step 2.1.3.5.3

Multiply $0$ by $0$ .

$\frac{0}{0}$

Step 2.1.3.5.4

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

Undefined

$\frac{0}{0}$

Step 2.1.3.6

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

Undefined

$\frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{0}{0}$

Step 2.2

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

Simplify each term.

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Step 2.3.2.1

Any root of $1$ is $1$ .

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

Step 2.3.2.2

Multiply $x$ by $1$ .

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

Step 2.3.3

Add $x$ and $x$ .

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

Step 2.3.4

By the Sum Rule, the derivative of $x - (2 x)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- (2 x)]$ .

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

Step 2.3.5

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

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

Step 2.3.6

Evaluate $\frac{d}{d x} [- (2 x)]$ .

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Step 2.3.6.1

Multiply $2$ by $- 1$ .

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

Step 2.3.6.2

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

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

Step 2.3.6.3

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

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

Step 2.3.6.4

Multiply $- 2$ by $1$ .

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

Step 2.3.7

Subtract $2$ from $1$ .

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

Step 2.3.8

Simplify each term.

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Step 2.3.8.1

Any root of $1$ is $1$ .

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

Step 2.3.8.2

Multiply $x$ by $1$ .

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

Step 2.3.9

Add $x$ and $x$ .

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

Step 2.3.10

Raise $x$ to the power of $1$ .

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

Step 2.3.11

Raise $x$ to the power of $1$ .

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

Step 2.3.12

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

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

Step 2.3.13

Add $1$ and $1$ .

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

Step 2.3.14

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

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

Step 2.3.15

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

$lim_{x \to 0} \frac{- 1}{2 (2 x)}$

Step 2.3.16

Multiply $2$ by $2$ .

$lim_{x \to 0} \frac{- 1}{4 x}$

Step 3

Since the function approaches $\infty$ from the left but $- \infty$ from the right, the limit does not exist.

$Does not exist$