Calculus Examples

$lim_{n \to 8} \frac{\sqrt{n + 1}}{\sqrt{n + 2}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $n$ approaches $8$ .

$\frac{lim_{n \to 8} \sqrt{n + 1}}{lim_{n \to 8} \sqrt{n + 2}}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{n \to 8} n + 1}}{lim_{n \to 8} \sqrt{n + 2}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $n$ approaches $8$ .

$\frac{\sqrt{lim_{n \to 8} n + lim_{n \to 8} 1}}{lim_{n \to 8} \sqrt{n + 2}}$

Step 4

Evaluate the limit of $1$ which is constant as $n$ approaches $8$ .

$\frac{\sqrt{lim_{n \to 8} n + 1}}{lim_{n \to 8} \sqrt{n + 2}}$

Step 5

Move the limit under the radical sign.

$\frac{\sqrt{lim_{n \to 8} n + 1}}{\sqrt{lim_{n \to 8} n + 2}}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $n$ approaches $8$ .

$\frac{\sqrt{lim_{n \to 8} n + 1}}{\sqrt{lim_{n \to 8} n + lim_{n \to 8} 2}}$

Step 7

Evaluate the limit of $2$ which is constant as $n$ approaches $8$ .

$\frac{\sqrt{lim_{n \to 8} n + 1}}{\sqrt{lim_{n \to 8} n + 2}}$

Step 8

Evaluate the limits by plugging in $8$ for all occurrences of $n$ .

Tap for more steps...

Step 8.1

Evaluate the limit of $n$ by plugging in $8$ for $n$ .

$\frac{\sqrt{8 + 1}}{\sqrt{lim_{n \to 8} n + 2}}$

Step 8.2

Evaluate the limit of $n$ by plugging in $8$ for $n$ .

$\frac{\sqrt{8 + 1}}{\sqrt{8 + 2}}$

Step 9

Simplify the answer.

Tap for more steps...

Step 9.1

Simplify the numerator.

Tap for more steps...

Step 9.1.1

Add $8$ and $1$ .

$\frac{\sqrt{9}}{\sqrt{8 + 2}}$

Step 9.1.2

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

$\frac{\sqrt{3^{2}}}{\sqrt{8 + 2}}$

Step 9.1.3

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

$\frac{3}{\sqrt{8 + 2}}$

Step 9.2

Add $8$ and $2$ .

$\frac{3}{\sqrt{10}}$

Step 9.3

Multiply $\frac{3}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{3}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 9.4

Combine and simplify the denominator.

Tap for more steps...

Step 9.4.1

Multiply $\frac{3}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{3 \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 9.4.2

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{3 \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Step 9.4.3

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{3 \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Step 9.4.4

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

$\frac{3 \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 9.4.5

Add $1$ and $1$ .

$\frac{3 \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 9.4.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Tap for more steps...

Step 9.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{3 \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 9.4.6.2

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

$\frac{3 \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 9.4.6.3

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

$\frac{3 \sqrt{10}}{10^{\frac{2}{2}}}$

Step 9.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.4.6.4.1

Cancel the common factor.

$\frac{3 \sqrt{10}}{10^{\frac{2}{2}}}$

Step 9.4.6.4.2

Rewrite the expression.

$\frac{3 \sqrt{10}}{10^{1}}$

Step 9.4.6.5

Evaluate the exponent.

$\frac{3 \sqrt{10}}{10}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{10}}{10}$

Decimal Form:

$0.94868329 \dots$