Calculus Examples

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

Step 1

Combine terms.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Write each expression with a common denominator of $(2 + x) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Reorder the factors of $(2 + x) \cdot 2$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

Simplify the limit argument.

Tap for more steps...

Step 2.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.2

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

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

Step 2.2

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

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

Step 3

Apply L'Hospital's rule.

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

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

Tap for more steps...

Step 3.1.2.1

Evaluate the limit of $2 - (2 + x)$ by plugging in $0$ for $x$ .

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

Step 3.1.2.2

Simplify each term.

Tap for more steps...

Step 3.1.2.2.1

Add $2$ and $0$ .

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

Step 3.1.2.2.2

Multiply $- 1$ by $2$ .

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

Step 3.1.2.3

Subtract $2$ from $2$ .

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

Step 3.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 3.1.3.1

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

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

Tap for more steps...

Step 3.1.3.4.1

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

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

Step 3.1.3.4.2

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

$\frac{1}{2} \cdot \frac{0}{(2 + 0) \cdot 0}$

Step 3.1.3.5

Simplify the answer.

Tap for more steps...

Step 3.1.3.5.1

Add $2$ and $0$ .

$\frac{1}{2} \cdot \frac{0}{2 \cdot 0}$

Step 3.1.3.5.2

Multiply $2$ by $0$ .

$\frac{1}{2} \cdot \frac{0}{0}$

Step 3.1.3.5.3

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

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

Step 3.1.3.6

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

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

Step 3.1.4

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

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

Tap for more steps...

Step 3.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (2 + x)$ with respect to $x$ is $- \frac{d}{d x} [2 + x]$ .

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

Step 3.3.4.2

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

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

Step 3.3.4.3

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

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

Step 3.3.4.4

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

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

Step 3.3.4.5

Add $0$ and $1$ .

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

Step 3.3.4.6

Multiply $- 1$ by $1$ .

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

Step 3.3.5

Subtract $1$ from $0$ .

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

Step 3.3.6

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 2 + x$ and $g (x) = x$ .

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

Step 3.3.7

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

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

Step 3.3.8

Multiply $2 + x$ by $1$ .

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

Step 3.3.9

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

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

Step 3.3.10

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

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

Step 3.3.11

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

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

Step 3.3.12

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

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

Step 3.3.13

Multiply $x$ by $1$ .

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

Step 3.3.14

Add $x$ and $x$ .

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

Step 3.3.15

Reorder terms.

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

Step 4

Evaluate the limit.

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 5

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

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

Step 6

Simplify the answer.

Tap for more steps...

Step 6.1

Simplify the denominator.

Tap for more steps...

Step 6.1.1

Multiply $2$ by $0$ .

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

Step 6.1.2

Add $0$ and $2$ .

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

Step 6.2

Move the negative in front of the fraction.

$\frac{1}{2} (- \frac{1}{2})$

Step 6.3

Multiply $\frac{1}{2} (- \frac{1}{2})$ .

Tap for more steps...

Step 6.3.1

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

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

Step 6.3.2

Multiply $2$ by $2$ .

$- \frac{1}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{4}$

Decimal Form:

$- 0.25$