Calculus Examples

$lim_{x \to 4} \frac{4 x - x^{2}}{2 - \sqrt{x}}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{x \to 4} 4 x - x^{2}}{lim_{x \to 4} 2 - \sqrt{x}}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} 4 x - lim_{x \to 4} x^{2}}{lim_{x \to 4} 2 - \sqrt{x}}$

Step 1.1.2.2

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 lim_{x \to 4} x - lim_{x \to 4} x^{2}}{lim_{x \to 4} 2 - \sqrt{x}}$

Step 1.1.2.3

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

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

Step 1.1.2.4

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 1.1.2.4.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{4 \cdot 4 - {(lim_{x \to 4} x)}^{2}}{lim_{x \to 4} 2 - \sqrt{x}}$

Step 1.1.2.4.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{4 \cdot 4 - 4^{2}}{lim_{x \to 4} 2 - \sqrt{x}}$

Step 1.1.2.5

Simplify the answer.

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Step 1.1.2.5.1

Simplify each term.

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Step 1.1.2.5.1.1

Multiply $4$ by $4$ .

$\frac{16 - 4^{2}}{lim_{x \to 4} 2 - \sqrt{x}}$

Step 1.1.2.5.1.2

Raise $4$ to the power of $2$ .

$\frac{16 - 1 \cdot 16}{lim_{x \to 4} 2 - \sqrt{x}}$

Step 1.1.2.5.1.3

Multiply $- 1$ by $16$ .

$\frac{16 - 16}{lim_{x \to 4} 2 - \sqrt{x}}$

Step 1.1.2.5.2

Subtract $16$ from $16$ .

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

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

Step 1.1.3.1.2

Evaluate the limit of $2$ which is constant as $x$ approaches $4$ .

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

Step 1.1.3.1.3

Move the limit under the radical sign.

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

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.1.3.3.1.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 1.1.3.3.1.3

Multiply $- 1$ by $2$ .

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

Step 1.1.3.3.2

Subtract $2$ from $2$ .

$\frac{0}{0}$

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.

$lim_{x \to 4} \frac{4 x - x^{2}}{2 - \sqrt{x}} = lim_{x \to 4} \frac{\frac{d}{d x} [4 x - x^{2}]}{\frac{d}{d x} [2 - \sqrt{x}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 4} \frac{\frac{d}{d x} [4 x - x^{2}]}{\frac{d}{d x} [2 - \sqrt{x}]}$

Step 1.3.2

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

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

Step 1.3.3

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 1.3.3.1

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

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

Step 1.3.3.2

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

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

Step 1.3.3.3

Multiply $4$ by $1$ .

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

Step 1.3.4

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

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Step 1.3.4.1

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

$lim_{x \to 4} \frac{4 - \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [2 - \sqrt{x}]}$

Step 1.3.4.2

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

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

Step 1.3.4.3

Multiply $2$ by $- 1$ .

$lim_{x \to 4} \frac{4 - 2 x}{\frac{d}{d x} [2 - \sqrt{x}]}$

Step 1.3.5

Reorder terms.

$lim_{x \to 4} \frac{- 2 x + 4}{\frac{d}{d x} [2 - \sqrt{x}]}$

Step 1.3.6

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

$lim_{x \to 4} \frac{- 2 x + 4}{\frac{d}{d x} [2] + \frac{d}{d x} [- \sqrt{x}]}$

Step 1.3.7

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

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

Step 1.3.8

Evaluate $\frac{d}{d x} [- \sqrt{x}]$ .

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Step 1.3.8.1

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

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

Step 1.3.8.2

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

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

Step 1.3.8.3

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

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

Step 1.3.8.4

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

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

Step 1.3.8.5

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

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

Step 1.3.8.6

Combine the numerators over the common denominator.

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

Step 1.3.8.7

Simplify the numerator.

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Step 1.3.8.7.1

Multiply $- 1$ by $2$ .

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

Step 1.3.8.7.2

Subtract $2$ from $1$ .

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

Step 1.3.8.8

Move the negative in front of the fraction.

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

Step 1.3.8.9

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

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

Step 1.3.8.10

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

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

Step 1.3.9

Subtract $\frac{1}{2 x^{\frac{1}{2}}}$ from $0$ .

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$lim_{x \to 4} (- 2 x + 4) (- (2 \sqrt{x}))$

Step 1.6

Multiply $2$ by $- 1$ .

$lim_{x \to 4} (- 2 x + 4) (- 2 \sqrt{x})$

Step 2

Evaluate the limit.

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Step 2.1

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

$- 2 lim_{x \to 4} (- 2 x + 4) \sqrt{x}$

Step 2.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $4$ .

$- 2 lim_{x \to 4} - 2 x + 4 \cdot lim_{x \to 4} \sqrt{x}$

Step 2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$- 2 (- lim_{x \to 4} 2 x + lim_{x \to 4} 4) \cdot lim_{x \to 4} \sqrt{x}$

Step 2.4

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

$- 2 (- 2 lim_{x \to 4} x + lim_{x \to 4} 4) \cdot lim_{x \to 4} \sqrt{x}$

Step 2.5

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

$- 2 (- 2 lim_{x \to 4} x + 4) \cdot lim_{x \to 4} \sqrt{x}$

Step 2.6

Move the limit under the radical sign.

$- 2 (- 2 lim_{x \to 4} x + 4) \cdot \sqrt{lim_{x \to 4} x}$

Step 3

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$- 2 (- 2 \cdot 4 + 4) \cdot \sqrt{lim_{x \to 4} x}$

Step 3.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$- 2 (- 2 \cdot 4 + 4) \cdot \sqrt{4}$

Step 4

Simplify the answer.

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Step 4.1

Multiply $- 2$ by $4$ .

$- 2 (- 8 + 4) \cdot \sqrt{4}$

Step 4.2

Add $- 8$ and $4$ .

$- 2 \cdot - 4 \cdot \sqrt{4}$

Step 4.3

Multiply $- 2$ by $- 4$ .

$8 \cdot \sqrt{4}$

Step 4.4

Rewrite $4$ as $2^{2}$ .

$8 \cdot \sqrt{2^{2}}$

Step 4.5

Pull terms out from under the radical, assuming positive real numbers.

$8 \cdot 2$

Step 4.6

Multiply $8$ by $2$ .

$16$