Precalculus Examples

$lim_{x \to 7} \frac{2 - \sqrt{x - 3}}{x^{2} - 49}$

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.

$\frac{lim_{x \to 7} 2 - \sqrt{x - 3}}{lim_{x \to 7} x^{2} - 49}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $7$ .

$\frac{lim_{x \to 7} 2 - lim_{x \to 7} \sqrt{x - 3}}{lim_{x \to 7} x^{2} - 49}$

Step 1.1.2.1.2

Evaluate the limit of $2$ which is constant as $x$ approaches $7$ .

$\frac{2 - lim_{x \to 7} \sqrt{x - 3}}{lim_{x \to 7} x^{2} - 49}$

Step 1.1.2.1.3

Move the limit under the radical sign.

$\frac{2 - \sqrt{lim_{x \to 7} x - 3}}{lim_{x \to 7} x^{2} - 49}$

Step 1.1.2.1.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $7$ .

$\frac{2 - \sqrt{lim_{x \to 7} x - lim_{x \to 7} 3}}{lim_{x \to 7} x^{2} - 49}$

Step 1.1.2.1.5

Evaluate the limit of $3$ which is constant as $x$ approaches $7$ .

$\frac{2 - \sqrt{lim_{x \to 7} x - 1 \cdot 3}}{lim_{x \to 7} x^{2} - 49}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$\frac{2 - \sqrt{7 - 1 \cdot 3}}{lim_{x \to 7} x^{2} - 49}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Multiply $- 1$ by $3$ .

$\frac{2 - \sqrt{7 - 3}}{lim_{x \to 7} x^{2} - 49}$

Step 1.1.2.3.1.2

Subtract $3$ from $7$ .

$\frac{2 - \sqrt{4}}{lim_{x \to 7} x^{2} - 49}$

Step 1.1.2.3.1.3

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

$\frac{2 - \sqrt{2^{2}}}{lim_{x \to 7} x^{2} - 49}$

Step 1.1.2.3.1.4

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

$\frac{2 - 1 \cdot 2}{lim_{x \to 7} x^{2} - 49}$

Step 1.1.2.3.1.5

Multiply $- 1$ by $2$ .

$\frac{2 - 2}{lim_{x \to 7} x^{2} - 49}$

Step 1.1.2.3.2

Subtract $2$ from $2$ .

$\frac{0}{lim_{x \to 7} x^{2} - 49}$

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 $7$ .

$\frac{0}{lim_{x \to 7} x^{2} - lim_{x \to 7} 49}$

Step 1.1.3.1.2

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

$\frac{0}{{(lim_{x \to 7} x)}^{2} - lim_{x \to 7} 49}$

Step 1.1.3.1.3

Evaluate the limit of $49$ which is constant as $x$ approaches $7$ .

$\frac{0}{{(lim_{x \to 7} x)}^{2} - 1 \cdot 49}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$\frac{0}{7^{2} - 1 \cdot 49}$

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

Raise $7$ to the power of $2$ .

$\frac{0}{49 - 1 \cdot 49}$

Step 1.1.3.3.1.2

Multiply $- 1$ by $49$ .

$\frac{0}{49 - 49}$

Step 1.1.3.3.2

Subtract $49$ from $49$ .

$\frac{0}{0}$

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.

$lim_{x \to 7} \frac{2 - \sqrt{x - 3}}{x^{2} - 49} = lim_{x \to 7} \frac{\frac{d}{d x} [2 - \sqrt{x - 3}]}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 7} \frac{\frac{d}{d x} [2 - \sqrt{x - 3}]}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.2

By the Sum Rule, the derivative of $2 - \sqrt{x - 3}$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- \sqrt{x - 3}]$ .

$lim_{x \to 7} \frac{\frac{d}{d x} [2] + \frac{d}{d x} [- \sqrt{x - 3}]}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.3

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

$lim_{x \to 7} \frac{0 + \frac{d}{d x} [- \sqrt{x - 3}]}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- \sqrt{x - 3}]$ .

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Step 1.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 3}$ as ${(x - 3)}^{\frac{1}{2}}$ .

$lim_{x \to 7} \frac{0 + \frac{d}{d x} [- {(x - 3)}^{\frac{1}{2}}]}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(x - 3)}^{\frac{1}{2}}$ with respect to $x$ is $- \frac{d}{d x} [{(x - 3)}^{\frac{1}{2}}]$ .

$lim_{x \to 7} \frac{0 - \frac{d}{d x} [{(x - 3)}^{\frac{1}{2}}]}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x - 3$ .

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Step 1.3.4.3.1

To apply the Chain Rule, set $u$ as $x - 3$ .

$lim_{x \to 7} \frac{0 - (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x - 3])}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4.3.2

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

$lim_{x \to 7} \frac{0 - (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x - 3])}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4.3.3

Replace all occurrences of $u$ with $x - 3$ .

$lim_{x \to 7} \frac{0 - (\frac{1}{2} {(x - 3)}^{\frac{1}{2} - 1} \frac{d}{d x} [x - 3])}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4.4

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$lim_{x \to 7} \frac{0 - (\frac{1}{2} {(x - 3)}^{\frac{1}{2} - 1} (\frac{d}{d x} [x] + \frac{d}{d x} [- 3]))}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4.5

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

$lim_{x \to 7} \frac{0 - (\frac{1}{2} {(x - 3)}^{\frac{1}{2} - 1} (1 + \frac{d}{d x} [- 3]))}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4.6

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$lim_{x \to 7} \frac{0 - (\frac{1}{2} {(x - 3)}^{\frac{1}{2} - 1} (1 + 0))}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4.7

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

$lim_{x \to 7} \frac{0 - (\frac{1}{2} {(x - 3)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (1 + 0))}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4.8

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

$lim_{x \to 7} \frac{0 - (\frac{1}{2} {(x - 3)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (1 + 0))}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4.9

Combine the numerators over the common denominator.

$lim_{x \to 7} \frac{0 - (\frac{1}{2} {(x - 3)}^{\frac{1 - 1 \cdot 2}{2}} (1 + 0))}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4.10

Simplify the numerator.

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Step 1.3.4.10.1

Multiply $- 1$ by $2$ .

$lim_{x \to 7} \frac{0 - (\frac{1}{2} {(x - 3)}^{\frac{1 - 2}{2}} (1 + 0))}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4.10.2

Subtract $2$ from $1$ .

$lim_{x \to 7} \frac{0 - (\frac{1}{2} {(x - 3)}^{\frac{- 1}{2}} (1 + 0))}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4.11

Move the negative in front of the fraction.

$lim_{x \to 7} \frac{0 - (\frac{1}{2} {(x - 3)}^{- \frac{1}{2}} (1 + 0))}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4.12

Add $1$ and $0$ .

$lim_{x \to 7} \frac{0 - (\frac{1}{2} {(x - 3)}^{- \frac{1}{2}} \cdot 1)}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4.13

Combine $\frac{1}{2}$ and ${(x - 3)}^{- \frac{1}{2}}$ .

$lim_{x \to 7} \frac{0 - (\frac{{(x - 3)}^{- \frac{1}{2}}}{2} \cdot 1)}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4.14

Multiply $\frac{{(x - 3)}^{- \frac{1}{2}}}{2}$ by $1$ .

$lim_{x \to 7} \frac{0 - \frac{{(x - 3)}^{- \frac{1}{2}}}{2}}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.4.15

Move ${(x - 3)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 7} \frac{0 - \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.5

Subtract $\frac{1}{2 {(x - 3)}^{\frac{1}{2}}}$ from $0$ .

$lim_{x \to 7} \frac{- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}}{\frac{d}{d x} [x^{2} - 49]}$

Step 1.3.6

By the Sum Rule, the derivative of $x^{2} - 49$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 49]$ .

$lim_{x \to 7} \frac{- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 49]}$

Step 1.3.7

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

$lim_{x \to 7} \frac{- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}}{2 x + \frac{d}{d x} [- 49]}$

Step 1.3.8

Since $- 49$ is constant with respect to $x$ , the derivative of $- 49$ with respect to $x$ is $0$ .

$lim_{x \to 7} \frac{- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}}{2 x + 0}$

Step 1.3.9

Add $2 x$ and $0$ .

$lim_{x \to 7} \frac{- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}}{2 x}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 7} - \frac{1}{2 {(x - 3)}^{\frac{1}{2}}} \cdot \frac{1}{2 x}$

Step 1.5

Rewrite ${(x - 3)}^{\frac{1}{2}}$ as $\sqrt{x - 3}$ .

$lim_{x \to 7} - \frac{1}{2 \sqrt{x - 3}} \cdot \frac{1}{2 x}$

Step 1.6

Combine factors.

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Step 1.6.1

Multiply $\frac{1}{2 x}$ by $\frac{1}{2 \sqrt{x - 3}}$ .

$lim_{x \to 7} - \frac{1}{2 x (2 \sqrt{x - 3})}$

Step 1.6.2

Multiply $2$ by $2$ .

$lim_{x \to 7} - \frac{1}{4 x \sqrt{x - 3}}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$- lim_{x \to 7} \frac{1}{4 x \sqrt{x - 3}}$

Step 2.2

Move the term $\frac{1}{4}$ outside of the limit because it is constant with respect to $x$ .

$- \frac{1}{4} lim_{x \to 7} \frac{1}{x \sqrt{x - 3}}$

Step 2.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $7$ .

$- \frac{1}{4} \cdot \frac{lim_{x \to 7} 1}{lim_{x \to 7} x \sqrt{x - 3}}$

Step 2.4

Evaluate the limit of $1$ which is constant as $x$ approaches $7$ .

$- \frac{1}{4} \cdot \frac{1}{lim_{x \to 7} x \sqrt{x - 3}}$

Step 2.5

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $7$ .

$- \frac{1}{4} \cdot \frac{1}{lim_{x \to 7} x \cdot lim_{x \to 7} \sqrt{x - 3}}$

Step 2.6

Move the limit under the radical sign.

$- \frac{1}{4} \cdot \frac{1}{lim_{x \to 7} x \cdot \sqrt{lim_{x \to 7} x - 3}}$

Step 2.7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $7$ .

$- \frac{1}{4} \cdot \frac{1}{lim_{x \to 7} x \cdot \sqrt{lim_{x \to 7} x - lim_{x \to 7} 3}}$

Step 2.8

Evaluate the limit of $3$ which is constant as $x$ approaches $7$ .

$- \frac{1}{4} \cdot \frac{1}{lim_{x \to 7} x \cdot \sqrt{lim_{x \to 7} x - 1 \cdot 3}}$

Step 3

Evaluate the limits by plugging in $7$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$- \frac{1}{4} \cdot \frac{1}{7 \cdot \sqrt{lim_{x \to 7} x - 1 \cdot 3}}$

Step 3.2

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$- \frac{1}{4} \cdot \frac{1}{7 \cdot \sqrt{7 - 1 \cdot 3}}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the denominator.

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Step 4.1.1

Multiply $- 1$ by $3$ .

$- \frac{1}{4} \cdot \frac{1}{7 \sqrt{7 - 3}}$

Step 4.1.2

Subtract $3$ from $7$ .

$- \frac{1}{4} \cdot \frac{1}{7 \sqrt{4}}$

Step 4.1.3

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

$- \frac{1}{4} \cdot \frac{1}{7 \sqrt{2^{2}}}$

Step 4.1.4

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

$- \frac{1}{4} \cdot \frac{1}{7 \cdot 2}$

Step 4.2

Multiply $7$ by $2$ .

$- \frac{1}{4} \cdot \frac{1}{14}$

Step 4.3

Multiply $- \frac{1}{4} \cdot \frac{1}{14}$ .

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Step 4.3.1

Multiply $\frac{1}{14}$ by $\frac{1}{4}$ .

$- \frac{1}{14 \cdot 4}$

Step 4.3.2

Multiply $14$ by $4$ .

$- \frac{1}{56}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{56}$

Decimal Form:

$- 0.01785714 \dots$