Calculus Examples

$lim_{x \to 0} \frac{- 2 x}{4 - {(2 + x)}^{2}}$

Step 1

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

$- 2 lim_{x \to 0} \frac{x}{4 - {(2 + x)}^{2}}$

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.

$- 2 \frac{lim_{x \to 0} x}{lim_{x \to 0} 4 - {(2 + x)}^{2}}$

Step 2.1.2

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

$- 2 \frac{0}{lim_{x \to 0} 4 - {(2 + x)}^{2}}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Evaluate the limit.

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Step 2.1.3.1.1

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

$- 2 \frac{0}{lim_{x \to 0} 4 - lim_{x \to 0} {(2 + x)}^{2}}$

Step 2.1.3.1.2

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

$- 2 \frac{0}{4 - lim_{x \to 0} {(2 + x)}^{2}}$

Step 2.1.3.1.3

Move the exponent $2$ from ${(2 + x)}^{2}$ outside the limit using the Limits Power Rule.

$- 2 \frac{0}{4 - {(lim_{x \to 0} 2 + x)}^{2}}$

Step 2.1.3.1.4

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

$- 2 \frac{0}{4 - {(lim_{x \to 0} 2 + lim_{x \to 0} x)}^{2}}$

Step 2.1.3.1.5

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

$- 2 \frac{0}{4 - {(2 + lim_{x \to 0} x)}^{2}}$

Step 2.1.3.2

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

$- 2 \frac{0}{4 - {(2 + 0)}^{2}}$

Step 2.1.3.3

Simplify the answer.

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Step 2.1.3.3.1

Simplify each term.

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Step 2.1.3.3.1.1

Add $2$ and $0$ .

$- 2 \frac{0}{4 - 2^{2}}$

Step 2.1.3.3.1.2

Raise $2$ to the power of $2$ .

$- 2 \frac{0}{4 - 1 \cdot 4}$

Step 2.1.3.3.1.3

Multiply $- 1$ by $4$ .

$- 2 \frac{0}{4 - 4}$

Step 2.1.3.3.2

Subtract $4$ from $4$ .

$- 2 (\frac{0}{0})$

Step 2.1.3.3.3

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

Undefined

$- 2 (\frac{0}{0})$

Step 2.1.3.4

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

Undefined

$- 2 (\frac{0}{0})$

Step 2.1.4

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

Undefined

$- 2 (\frac{0}{0})$

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.

$lim_{x \to 0} \frac{x}{4 - {(2 + x)}^{2}} = lim_{x \to 0} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [4 - {(2 + x)}^{2}]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$- 2 lim_{x \to 0} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [4 - {(2 + x)}^{2}]}$

Step 2.3.2

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

$- 2 lim_{x \to 0} \frac{1}{\frac{d}{d x} [4 - {(2 + x)}^{2}]}$

Step 2.3.3

Rewrite ${(2 + x)}^{2}$ as $(2 + x) (2 + x)$ .

$- 2 lim_{x \to 0} \frac{1}{\frac{d}{d x} [4 - (2 + x) (2 + x)]}$

Step 2.3.4

Expand $(2 + x) (2 + x)$ using the FOIL Method.

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Step 2.3.4.1

Apply the distributive property.

$- 2 lim_{x \to 0} \frac{1}{\frac{d}{d x} [4 - (2 (2 + x) + x (2 + x))]}$

Step 2.3.4.2

Apply the distributive property.

$- 2 lim_{x \to 0} \frac{1}{\frac{d}{d x} [4 - (2 \cdot 2 + 2 x + x (2 + x))]}$

Step 2.3.4.3

Apply the distributive property.

$- 2 lim_{x \to 0} \frac{1}{\frac{d}{d x} [4 - (2 \cdot 2 + 2 x + x \cdot 2 + x \cdot x)]}$

Step 2.3.5

Simplify and combine like terms.

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Step 2.3.5.1

Simplify each term.

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Step 2.3.5.1.1

Multiply $2$ by $2$ .

$- 2 lim_{x \to 0} \frac{1}{\frac{d}{d x} [4 - (4 + 2 x + x \cdot 2 + x \cdot x)]}$

Step 2.3.5.1.2

Move $2$ to the left of $x$ .

$- 2 lim_{x \to 0} \frac{1}{\frac{d}{d x} [4 - (4 + 2 x + 2 \cdot x + x \cdot x)]}$

Step 2.3.5.1.3

Multiply $x$ by $x$ .

$- 2 lim_{x \to 0} \frac{1}{\frac{d}{d x} [4 - (4 + 2 x + 2 x + x^{2})]}$

Step 2.3.5.2

Add $2 x$ and $2 x$ .

$- 2 lim_{x \to 0} \frac{1}{\frac{d}{d x} [4 - (4 + 4 x + x^{2})]}$

Step 2.3.6

By the Sum Rule, the derivative of $4 - (4 + 4 x + x^{2})$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- (4 + 4 x + x^{2})]$ .

$- 2 lim_{x \to 0} \frac{1}{\frac{d}{d x} [4] + \frac{d}{d x} [- (4 + 4 x + x^{2})]}$

Step 2.3.7

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$- 2 lim_{x \to 0} \frac{1}{0 + \frac{d}{d x} [- (4 + 4 x + x^{2})]}$

Step 2.3.8

Evaluate $\frac{d}{d x} [- (4 + 4 x + x^{2})]$ .

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Step 2.3.8.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (4 + 4 x + x^{2})$ with respect to $x$ is $- \frac{d}{d x} [4 + 4 x + x^{2}]$ .

$- 2 lim_{x \to 0} \frac{1}{0 - \frac{d}{d x} [4 + 4 x + x^{2}]}$

Step 2.3.8.2

By the Sum Rule, the derivative of $4 + 4 x + x^{2}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [4 x] + \frac{d}{d x} [x^{2}]$ .

$- 2 lim_{x \to 0} \frac{1}{0 - (\frac{d}{d x} [4] + \frac{d}{d x} [4 x] + \frac{d}{d x} [x^{2}])}$

Step 2.3.8.3

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$- 2 lim_{x \to 0} \frac{1}{0 - (0 + \frac{d}{d x} [4 x] + \frac{d}{d x} [x^{2}])}$

Step 2.3.8.4

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$- 2 lim_{x \to 0} \frac{1}{0 - (0 + 4 \frac{d}{d x} [x] + \frac{d}{d x} [x^{2}])}$

Step 2.3.8.5

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

$- 2 lim_{x \to 0} \frac{1}{0 - (0 + 4 \cdot 1 + \frac{d}{d x} [x^{2}])}$

Step 2.3.8.6

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

$- 2 lim_{x \to 0} \frac{1}{0 - (0 + 4 \cdot 1 + 2 x)}$

Step 2.3.8.7

Multiply $4$ by $1$ .

$- 2 lim_{x \to 0} \frac{1}{0 - (0 + 4 + 2 x)}$

Step 2.3.8.8

Add $0$ and $4$ .

$- 2 lim_{x \to 0} \frac{1}{0 - (4 + 2 x)}$

Step 2.3.9

Simplify.

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Step 2.3.9.1

Apply the distributive property.

$- 2 lim_{x \to 0} \frac{1}{0 - 1 \cdot 4 - 1 \cdot 2 x}$

Step 2.3.9.2

Combine terms.

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Step 2.3.9.2.1

Multiply $- 1$ by $4$ .

$- 2 lim_{x \to 0} \frac{1}{0 - 4 - 1 \cdot 2 x}$

Step 2.3.9.2.2

Multiply $2$ by $- 1$ .

$- 2 lim_{x \to 0} \frac{1}{0 - 4 - 2 x}$

Step 2.3.9.2.3

Subtract $- (- 4 - 2 x)$ from $0$ .

$- 2 lim_{x \to 0} \frac{1}{- 4 - 2 x}$

Step 2.3.9.3

Reorder terms.

$- 2 lim_{x \to 0} \frac{1}{- 2 x - 4}$

Step 3

Evaluate the limit.

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Step 3.1

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

$- 2 \frac{lim_{x \to 0} 1}{lim_{x \to 0} - 2 x - 4}$

Step 3.2

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

$- 2 \frac{1}{lim_{x \to 0} - 2 x - 4}$

Step 3.3

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

$- 2 \frac{1}{- lim_{x \to 0} 2 x - lim_{x \to 0} 4}$

Step 3.4

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

$- 2 \frac{1}{- 2 lim_{x \to 0} x - lim_{x \to 0} 4}$

Step 3.5

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

$- 2 \frac{1}{- 2 lim_{x \to 0} x - 1 \cdot 4}$

Step 4

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

$- 2 \frac{1}{- 2 \cdot 0 - 1 \cdot 4}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the denominator.

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Step 5.1.1

Multiply $- 2$ by $0$ .

$- 2 \frac{1}{0 - 1 \cdot 4}$

Step 5.1.2

Multiply $- 1$ by $4$ .

$- 2 \frac{1}{0 - 4}$

Step 5.1.3

Subtract $4$ from $0$ .

$- 2 (\frac{1}{- 4})$

Step 5.2

Cancel the common factor of $2$ .

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Step 5.2.1

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{1}{- 4}$

Step 5.2.2

Factor $2$ out of $- 4$ .

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

Step 5.2.3

Cancel the common factor.

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

Step 5.2.4

Rewrite the expression.

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

Step 5.3

Move the negative in front of the fraction.

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

Step 5.4

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

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Step 5.4.1

Multiply $- 1$ by $- 1$ .

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

Step 5.4.2

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

$\frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$