Calculus Examples

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

Step 1

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

$\frac{1}{\sqrt{x + h + 1}} - \frac{1}{\sqrt{x + 1}}$

Step 2

Simplify the answer.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Multiply $\frac{1}{\sqrt{x + h + 1}}$ by $\frac{\sqrt{x + h + 1}}{\sqrt{x + h + 1}}$ .

$\frac{1}{\sqrt{x + h + 1}} \cdot \frac{\sqrt{x + h + 1}}{\sqrt{x + h + 1}} - \frac{1}{\sqrt{x + 1}}$

Step 2.1.2

Combine and simplify the denominator.

Tap for more steps...

Step 2.1.2.1

Multiply $\frac{1}{\sqrt{x + h + 1}}$ by $\frac{\sqrt{x + h + 1}}{\sqrt{x + h + 1}}$ .

$\frac{\sqrt{x + h + 1}}{\sqrt{x + h + 1} \sqrt{x + h + 1}} - \frac{1}{\sqrt{x + 1}}$

Step 2.1.2.2

Raise $\sqrt{x + h + 1}$ to the power of $1$ .

$\frac{\sqrt{x + h + 1}}{{\sqrt{x + h + 1}}^{1} \sqrt{x + h + 1}} - \frac{1}{\sqrt{x + 1}}$

Step 2.1.2.3

Raise $\sqrt{x + h + 1}$ to the power of $1$ .

$\frac{\sqrt{x + h + 1}}{{\sqrt{x + h + 1}}^{1} {\sqrt{x + h + 1}}^{1}} - \frac{1}{\sqrt{x + 1}}$

Step 2.1.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{x + h + 1}}{{\sqrt{x + h + 1}}^{1 + 1}} - \frac{1}{\sqrt{x + 1}}$

Step 2.1.2.5

Add $1$ and $1$ .

$\frac{\sqrt{x + h + 1}}{{\sqrt{x + h + 1}}^{2}} - \frac{1}{\sqrt{x + 1}}$

Step 2.1.2.6

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

Tap for more steps...

Step 2.1.2.6.1

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

$\frac{\sqrt{x + h + 1}}{{({(x + h + 1)}^{\frac{1}{2}})}^{2}} - \frac{1}{\sqrt{x + 1}}$

Step 2.1.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{x + h + 1}}{{(x + h + 1)}^{\frac{1}{2} \cdot 2}} - \frac{1}{\sqrt{x + 1}}$

Step 2.1.2.6.3

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

$\frac{\sqrt{x + h + 1}}{{(x + h + 1)}^{\frac{2}{2}}} - \frac{1}{\sqrt{x + 1}}$

Step 2.1.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.2.6.4.1

Cancel the common factor.

$\frac{\sqrt{x + h + 1}}{{(x + h + 1)}^{\frac{2}{2}}} - \frac{1}{\sqrt{x + 1}}$

Step 2.1.2.6.4.2

Rewrite the expression.

$\frac{\sqrt{x + h + 1}}{{(x + h + 1)}^{1}} - \frac{1}{\sqrt{x + 1}}$

Step 2.1.2.6.5

Simplify.

$\frac{\sqrt{x + h + 1}}{x + h + 1} - \frac{1}{\sqrt{x + 1}}$

Step 2.1.3

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

$\frac{\sqrt{x + h + 1}}{x + h + 1} - (\frac{1}{\sqrt{x + 1}} \cdot \frac{\sqrt{x + 1}}{\sqrt{x + 1}})$

Step 2.1.4

Combine and simplify the denominator.

Tap for more steps...

Step 2.1.4.1

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

$\frac{\sqrt{x + h + 1}}{x + h + 1} - \frac{\sqrt{x + 1}}{\sqrt{x + 1} \sqrt{x + 1}}$

Step 2.1.4.2

Raise $\sqrt{x + 1}$ to the power of $1$ .

$\frac{\sqrt{x + h + 1}}{x + h + 1} - \frac{\sqrt{x + 1}}{{\sqrt{x + 1}}^{1} \sqrt{x + 1}}$

Step 2.1.4.3

Raise $\sqrt{x + 1}$ to the power of $1$ .

$\frac{\sqrt{x + h + 1}}{x + h + 1} - \frac{\sqrt{x + 1}}{{\sqrt{x + 1}}^{1} {\sqrt{x + 1}}^{1}}$

Step 2.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{x + h + 1}}{x + h + 1} - \frac{\sqrt{x + 1}}{{\sqrt{x + 1}}^{1 + 1}}$

Step 2.1.4.5

Add $1$ and $1$ .

$\frac{\sqrt{x + h + 1}}{x + h + 1} - \frac{\sqrt{x + 1}}{{\sqrt{x + 1}}^{2}}$

Step 2.1.4.6

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

Tap for more steps...

Step 2.1.4.6.1

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

$\frac{\sqrt{x + h + 1}}{x + h + 1} - \frac{\sqrt{x + 1}}{{({(x + 1)}^{\frac{1}{2}})}^{2}}$

Step 2.1.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{x + h + 1}}{x + h + 1} - \frac{\sqrt{x + 1}}{{(x + 1)}^{\frac{1}{2} \cdot 2}}$

Step 2.1.4.6.3

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

$\frac{\sqrt{x + h + 1}}{x + h + 1} - \frac{\sqrt{x + 1}}{{(x + 1)}^{\frac{2}{2}}}$

Step 2.1.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.4.6.4.1

Cancel the common factor.

$\frac{\sqrt{x + h + 1}}{x + h + 1} - \frac{\sqrt{x + 1}}{{(x + 1)}^{\frac{2}{2}}}$

Step 2.1.4.6.4.2

Rewrite the expression.

$\frac{\sqrt{x + h + 1}}{x + h + 1} - \frac{\sqrt{x + 1}}{{(x + 1)}^{1}}$

Step 2.1.4.6.5

Simplify.

$\frac{\sqrt{x + h + 1}}{x + h + 1} - \frac{\sqrt{x + 1}}{x + 1}$

Step 2.2

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

$\frac{\sqrt{x + h + 1}}{x + h + 1} \cdot \frac{x + 1}{x + 1} - \frac{\sqrt{x + 1}}{x + 1}$

Step 2.3

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

$\frac{\sqrt{x + h + 1}}{x + h + 1} \cdot \frac{x + 1}{x + 1} - \frac{\sqrt{x + 1}}{x + 1} \cdot \frac{x + h + 1}{x + h + 1}$

Step 2.4

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

Tap for more steps...

Step 2.4.1

Multiply $\frac{\sqrt{x + h + 1}}{x + h + 1}$ by $\frac{x + 1}{x + 1}$ .

$\frac{\sqrt{x + h + 1} (x + 1)}{(x + h + 1) (x + 1)} - \frac{\sqrt{x + 1}}{x + 1} \cdot \frac{x + h + 1}{x + h + 1}$

Step 2.4.2

Multiply $\frac{\sqrt{x + 1}}{x + 1}$ by $\frac{x + h + 1}{x + h + 1}$ .

$\frac{\sqrt{x + h + 1} (x + 1)}{(x + h + 1) (x + 1)} - \frac{\sqrt{x + 1} (x + h + 1)}{(x + 1) (x + h + 1)}$

Step 2.4.3

Reorder the factors of $(x + h + 1) (x + 1)$ .

$\frac{\sqrt{x + h + 1} (x + 1)}{(x + 1) (x + h + 1)} - \frac{\sqrt{x + 1} (x + h + 1)}{(x + 1) (x + h + 1)}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{\sqrt{x + h + 1} (x + 1) - \sqrt{x + 1} (x + h + 1)}{(x + 1) (x + h + 1)}$

Step 2.6

Simplify the numerator.

Tap for more steps...

Step 2.6.1

Apply the distributive property.

$\frac{\sqrt{x + h + 1} x + \sqrt{x + h + 1} \cdot 1 - \sqrt{x + 1} (x + h + 1)}{(x + 1) (x + h + 1)}$

Step 2.6.2

Multiply $\sqrt{x + h + 1}$ by $1$ .

$\frac{\sqrt{x + h + 1} x + \sqrt{x + h + 1} - \sqrt{x + 1} (x + h + 1)}{(x + 1) (x + h + 1)}$

Step 2.6.3

Apply the distributive property.

$\frac{\sqrt{x + h + 1} x + \sqrt{x + h + 1} - \sqrt{x + 1} x - \sqrt{x + 1} h - \sqrt{x + 1} \cdot 1}{(x + 1) (x + h + 1)}$

Step 2.6.4

Multiply $- 1$ by $1$ .

$\frac{\sqrt{x + h + 1} x + \sqrt{x + h + 1} - \sqrt{x + 1} x - \sqrt{x + 1} h - \sqrt{x + 1}}{(x + 1) (x + h + 1)}$