Calculus Examples

$lim_{h \to 0} \frac{\frac{2}{x + h} - \frac{2}{x}}{h}$

Step 1

Combine terms.

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Step 1.1

To write $\frac{2}{x + h}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$lim_{h \to 0} \frac{\frac{2}{x + h} \cdot \frac{x}{x} - \frac{2}{x}}{h}$

Step 1.2

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

$lim_{h \to 0} \frac{\frac{2}{x + h} \cdot \frac{x}{x} - \frac{2}{x} \cdot \frac{x + h}{x + h}}{h}$

Step 1.3

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

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Step 1.3.1

Multiply $\frac{2}{x + h}$ by $\frac{x}{x}$ .

$lim_{h \to 0} \frac{\frac{2 x}{(x + h) x} - \frac{2}{x} \cdot \frac{x + h}{x + h}}{h}$

Step 1.3.2

Multiply $\frac{2}{x}$ by $\frac{x + h}{x + h}$ .

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

Step 1.3.3

Reorder the factors of $(x + h) x$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 2

Evaluate the limit.

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Step 2.1

Simplify the limit argument.

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Step 2.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.2

Multiply $\frac{2 x - 2 (x + h)}{x (x + h)}$ by $\frac{1}{h}$ .

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

Step 2.2

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

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

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Evaluate the limit.

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Step 3.1.2.1.1

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

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

Step 3.1.2.1.2

Evaluate the limit of $2 x$ which is constant as $h$ approaches $0$ .

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

Step 3.1.2.1.3

Move the term $2$ outside of the limit because it is constant with respect to $h$ .

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

Step 3.1.2.1.4

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

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

Step 3.1.2.1.5

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

Combine the opposite terms in $2 x - 2 (x + 0)$ .

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Step 3.1.2.3.1

Add $x$ and $0$ .

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

Step 3.1.2.3.2

Subtract $2 x$ from $2 x$ .

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

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

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

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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Step 3.1.3.4.1

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

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

Step 3.1.3.4.2

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

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

Step 3.1.3.5

Simplify the answer.

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Step 3.1.3.5.1

Add $x$ and $0$ .

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

Step 3.1.3.5.2

Multiply $x$ by $0$ .

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

Step 3.1.3.5.3

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

Undefined

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

Step 3.1.3.6

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

Undefined

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

Step 3.1.4

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

Undefined

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

Step 3.2

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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

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

Step 3.3.3

Since $2 x$ is constant with respect to $h$ , the derivative of $2 x$ with respect to $h$ is $0$ .

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

Step 3.3.4

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

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Step 3.3.4.1

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

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

Step 3.3.4.2

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

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

Step 3.3.4.3

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

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

Step 3.3.4.4

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

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

Step 3.3.4.5

Add $0$ and $1$ .

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

Step 3.3.4.6

Multiply $- 2$ by $1$ .

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

Step 3.3.5

Subtract $2$ from $0$ .

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

Step 3.3.6

Differentiate using the Product Rule which states that $\frac{d}{d h} [f (h) g (h)]$ is $f (h) \frac{d}{d h} [g (h)] + g (h) \frac{d}{d h} [f (h)]$ where $f (h) = x + h$ and $g (h) = h$ .

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

Step 3.3.7

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

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

Step 3.3.8

Multiply $x + h$ by $1$ .

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

Step 3.3.9

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

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

Step 3.3.10

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

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

Step 3.3.11

Add $0$ and $\frac{d}{d h} [h]$ .

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

Step 3.3.12

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

$\frac{1}{x} lim_{h \to 0} \frac{- 2}{x + h + h \cdot 1}$

Step 3.3.13

Multiply $h$ by $1$ .

$\frac{1}{x} lim_{h \to 0} \frac{- 2}{x + h + h}$

Step 3.3.14

Add $h$ and $h$ .

$\frac{1}{x} lim_{h \to 0} \frac{- 2}{x + 2 h}$

Step 3.3.15

Reorder terms.

$\frac{1}{x} lim_{h \to 0} \frac{- 2}{2 h + x}$

Step 4

Evaluate the limit.

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Step 4.1

Move the term $- 2$ outside of the limit because it is constant with respect to $h$ .

$\frac{1}{x} \cdot - 2 lim_{h \to 0} \frac{1}{2 h + x}$

Step 4.2

Split the limit using the Limits Quotient Rule on the limit as $h$ approaches $0$ .

$\frac{1}{x} \cdot - 2 \frac{lim_{h \to 0} 1}{lim_{h \to 0} 2 h + x}$

Step 4.3

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

$\frac{1}{x} \cdot - 2 \frac{1}{lim_{h \to 0} 2 h + x}$

Step 4.4

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

$\frac{1}{x} \cdot - 2 \frac{1}{lim_{h \to 0} 2 h + lim_{h \to 0} x}$

Step 4.5

Move the term $2$ outside of the limit because it is constant with respect to $h$ .

$\frac{1}{x} \cdot - 2 \frac{1}{2 lim_{h \to 0} h + lim_{h \to 0} x}$

Step 4.6

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

$\frac{1}{x} \cdot - 2 \frac{1}{2 lim_{h \to 0} h + x}$

Step 5

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

$\frac{1}{x} \cdot - 2 \frac{1}{2 \cdot 0 + x}$

Step 6

Simplify the answer.

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Step 6.1

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

$\frac{- 2}{x} \cdot \frac{1}{2 \cdot 0 + x}$

Step 6.2

Move the negative in front of the fraction.

$- \frac{2}{x} \cdot \frac{1}{2 \cdot 0 + x}$

Step 6.3

Simplify the denominator.

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Step 6.3.1

Multiply $2$ by $0$ .

$- \frac{2}{x} \cdot \frac{1}{0 + x}$

Step 6.3.2

Add $0$ and $x$ .

$- \frac{2}{x} \cdot \frac{1}{x}$

Step 6.4

Multiply $- \frac{2}{x} \cdot \frac{1}{x}$ .

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Step 6.4.1

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

$- \frac{2}{x \cdot x}$

Step 6.4.2

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

$- \frac{2}{x^{1} x}$

Step 6.4.3

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

$- \frac{2}{x^{1} x^{1}}$

Step 6.4.4

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

$- \frac{2}{x^{1 + 1}}$

Step 6.4.5

Add $1$ and $1$ .

$- \frac{2}{x^{2}}$