Calculus Examples

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

Step 1

Apply L'Hospital's rule.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

Evaluate the limit of the numerator.

Tap for more steps...

Step 1.1.2.1

Evaluate the limit.

Tap for more steps...

Step 1.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

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

Step 1.1.2.1.2

Move the limit under the radical sign.

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

Step 1.1.2.1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

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

Step 1.1.2.1.4

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

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

Step 1.1.2.1.5

Evaluate the limit of $3$ which is constant as $x$ approaches $2$ .

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

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

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

Step 1.1.2.3

Simplify the answer.

Tap for more steps...

Step 1.1.2.3.1

Simplify each term.

Tap for more steps...

Step 1.1.2.3.1.1

Add $2$ and $7$ .

$\frac{\sqrt{9} - 1 \cdot 3}{lim_{x \to 2} 2 - x}$

Step 1.1.2.3.1.2

Rewrite $9$ as $3^{2}$ .

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

Step 1.1.2.3.1.3

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

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

Step 1.1.2.3.1.4

Multiply $- 1$ by $3$ .

$\frac{3 - 3}{lim_{x \to 2} 2 - x}$

Step 1.1.2.3.2

Subtract $3$ from $3$ .

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

Step 1.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 1.1.3.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Simplify the expression.

Tap for more steps...

Step 1.1.3.3.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

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

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

By the Sum Rule, the derivative of $\sqrt{x + 7} - 3$ with respect to $x$ is $\frac{d}{d x} [\sqrt{x + 7}] + \frac{d}{d x} [- 3]$ .

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

Step 1.3.3

Evaluate $\frac{d}{d x} [\sqrt{x + 7}]$ .

Tap for more steps...

Step 1.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 7}$ as ${(x + 7)}^{\frac{1}{2}}$ .

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

Step 1.3.3.2

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 $g (x) = x + 7$ .

Tap for more steps...

Step 1.3.3.2.1

To apply the Chain Rule, set $u$ as $x + 7$ .

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

Step 1.3.3.2.2

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

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

Step 1.3.3.2.3

Replace all occurrences of $u$ with $x + 7$ .

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

Step 1.3.3.3

By the Sum Rule, the derivative of $x + 7$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [7]$ .

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

Step 1.3.3.4

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

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

Step 1.3.3.5

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

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

Step 1.3.3.6

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

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

Step 1.3.3.7

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

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

Step 1.3.3.8

Combine the numerators over the common denominator.

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

Step 1.3.3.9

Simplify the numerator.

Tap for more steps...

Step 1.3.3.9.1

Multiply $- 1$ by $2$ .

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

Step 1.3.3.9.2

Subtract $2$ from $1$ .

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

Step 1.3.3.10

Move the negative in front of the fraction.

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

Step 1.3.3.11

Add $1$ and $0$ .

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

Step 1.3.3.12

Combine $\frac{1}{2}$ and ${(x + 7)}^{- \frac{1}{2}}$ .

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

Step 1.3.3.13

Multiply $\frac{{(x + 7)}^{- \frac{1}{2}}}{2}$ by $1$ .

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

Step 1.3.3.14

Move ${(x + 7)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.3.4

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

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

Step 1.3.5

Add $\frac{1}{2 {(x + 7)}^{\frac{1}{2}}}$ and $0$ .

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

Step 1.3.6

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

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

Step 1.3.7

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

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

Step 1.3.8

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

Tap for more steps...

Step 1.3.8.1

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

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

Step 1.3.8.2

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

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

Step 1.3.8.3

Multiply $- 1$ by $1$ .

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

Step 1.3.9

Subtract $1$ from $0$ .

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5

Rewrite ${(x + 7)}^{\frac{1}{2}}$ as $\sqrt{x + 7}$ .

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

Step 1.6

Combine factors.

Tap for more steps...

Step 1.6.1

Multiply $\frac{1}{2 \sqrt{x + 7}}$ by $\frac{1}{- 1}$ .

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

Step 1.6.2

Multiply $- 1$ by $2$ .

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

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

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

Step 2.3

Evaluate the limit of $1$ which is constant as $x$ approaches $2$ .

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

Step 2.4

Move the limit under the radical sign.

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

Step 2.5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

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

Step 2.6

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

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

Step 3

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{1}{- 2} \cdot \frac{1}{\sqrt{2 + 7}}$

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

Move the negative in front of the fraction.

$- \frac{1}{2} \cdot \frac{1}{\sqrt{2 + 7}}$

Step 4.2

Simplify the denominator.

Tap for more steps...

Step 4.2.1

Add $2$ and $7$ .

$- \frac{1}{2} \cdot \frac{1}{\sqrt{9}}$

Step 4.2.2

Rewrite $9$ as $3^{2}$ .

$- \frac{1}{2} \cdot \frac{1}{\sqrt{3^{2}}}$

Step 4.2.3

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

$- \frac{1}{2} \cdot \frac{1}{3}$

Step 4.3

Multiply $- \frac{1}{2} \cdot \frac{1}{3}$ .

Tap for more steps...

Step 4.3.1

Multiply $\frac{1}{3}$ by $\frac{1}{2}$ .

$- \frac{1}{3 \cdot 2}$

Step 4.3.2

Multiply $3$ by $2$ .

$- \frac{1}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{6}$

Decimal Form:

$- 0.1\overline{6}$