Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the limit under the radical sign.

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

Step 4

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

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

Step 5

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

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

Step 6

Move the limit under the radical sign.

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

Step 7

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

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

Step 8

Move the limit under the radical sign.

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

Step 9

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

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

Step 10

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

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

Step 11

Move the limit under the radical sign.

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

Step 12

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

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

Step 13

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

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

Step 14

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

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Step 14.1

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

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

Step 14.2

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

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

Step 14.3

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

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

Step 14.4

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

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

Step 15

Simplify the answer.

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Step 15.1

Simplify the numerator.

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Step 15.1.1

Add $8$ and $1$ .

$\frac{\sqrt{9} - \sqrt{8}}{\sqrt{8 + 2} - \sqrt{8 + 1}}$

Step 15.1.2

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

$\frac{\sqrt{3^{2}} - \sqrt{8}}{\sqrt{8 + 2} - \sqrt{8 + 1}}$

Step 15.1.3

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

$\frac{3 - \sqrt{8}}{\sqrt{8 + 2} - \sqrt{8 + 1}}$

Step 15.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 15.1.4.1

Factor $4$ out of $8$ .

$\frac{3 - \sqrt{4 (2)}}{\sqrt{8 + 2} - \sqrt{8 + 1}}$

Step 15.1.4.2

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

$\frac{3 - \sqrt{2^{2} \cdot 2}}{\sqrt{8 + 2} - \sqrt{8 + 1}}$

Step 15.1.5

Pull terms out from under the radical.

$\frac{3 - (2 \sqrt{2})}{\sqrt{8 + 2} - \sqrt{8 + 1}}$

Step 15.1.6

Multiply $2$ by $- 1$ .

$\frac{3 - 2 \sqrt{2}}{\sqrt{8 + 2} - \sqrt{8 + 1}}$

Step 15.2

Simplify the denominator.

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Step 15.2.1

Add $8$ and $2$ .

$\frac{3 - 2 \sqrt{2}}{\sqrt{10} - \sqrt{8 + 1}}$

Step 15.2.2

Add $8$ and $1$ .

$\frac{3 - 2 \sqrt{2}}{\sqrt{10} - \sqrt{9}}$

Step 15.2.3

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

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

Step 15.2.4

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

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

Step 15.2.5

Multiply $- 1$ by $3$ .

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

Step 15.3

Multiply $\frac{3 - 2 \sqrt{2}}{\sqrt{10} - 3}$ by $\frac{\sqrt{10} + 3}{\sqrt{10} + 3}$ .

$\frac{3 - 2 \sqrt{2}}{\sqrt{10} - 3} \cdot \frac{\sqrt{10} + 3}{\sqrt{10} + 3}$

Step 15.4

Multiply $\frac{3 - 2 \sqrt{2}}{\sqrt{10} - 3}$ by $\frac{\sqrt{10} + 3}{\sqrt{10} + 3}$ .

$\frac{(3 - 2 \sqrt{2}) (\sqrt{10} + 3)}{(\sqrt{10} - 3) (\sqrt{10} + 3)}$

Step 15.5

Expand the denominator using the FOIL method.

$\frac{(3 - 2 \sqrt{2}) (\sqrt{10} + 3)}{{\sqrt{10}}^{2} + \sqrt{10} \cdot 3 - 3 \sqrt{10} - 9}$

Step 15.6

Simplify.

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

Step 15.7

Divide $(3 - 2 \sqrt{2}) (\sqrt{10} + 3)$ by $1$ .

$(3 - 2 \sqrt{2}) (\sqrt{10} + 3)$

Step 15.8

Expand $(3 - 2 \sqrt{2}) (\sqrt{10} + 3)$ using the FOIL Method.

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Step 15.8.1

Apply the distributive property.

$3 (\sqrt{10} + 3) - 2 \sqrt{2} (\sqrt{10} + 3)$

Step 15.8.2

Apply the distributive property.

$3 \sqrt{10} + 3 \cdot 3 - 2 \sqrt{2} (\sqrt{10} + 3)$

Step 15.8.3

Apply the distributive property.

$3 \sqrt{10} + 3 \cdot 3 - 2 \sqrt{2} \sqrt{10} - 2 \sqrt{2} \cdot 3$

Step 15.9

Simplify each term.

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Step 15.9.1

Multiply $3$ by $3$ .

$3 \sqrt{10} + 9 - 2 \sqrt{2} \sqrt{10} - 2 \sqrt{2} \cdot 3$

Step 15.9.2

Multiply $- 2 \sqrt{2} \sqrt{10}$ .

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Step 15.9.2.1

Combine using the product rule for radicals.

$3 \sqrt{10} + 9 - 2 \sqrt{10 \cdot 2} - 2 \sqrt{2} \cdot 3$

Step 15.9.2.2

Multiply $10$ by $2$ .

$3 \sqrt{10} + 9 - 2 \sqrt{20} - 2 \sqrt{2} \cdot 3$

Step 15.9.3

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 15.9.3.1

Factor $4$ out of $20$ .

$3 \sqrt{10} + 9 - 2 \sqrt{4 (5)} - 2 \sqrt{2} \cdot 3$

Step 15.9.3.2

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

$3 \sqrt{10} + 9 - 2 \sqrt{2^{2} \cdot 5} - 2 \sqrt{2} \cdot 3$

Step 15.9.4

Pull terms out from under the radical.

$3 \sqrt{10} + 9 - 2 (2 \sqrt{5}) - 2 \sqrt{2} \cdot 3$

Step 15.9.5

Multiply $2$ by $- 2$ .

$3 \sqrt{10} + 9 - 4 \sqrt{5} - 2 \sqrt{2} \cdot 3$

Step 15.9.6

Multiply $3$ by $- 2$ .

$3 \sqrt{10} + 9 - 4 \sqrt{5} - 6 \sqrt{2}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{10} + 9 - 4 \sqrt{5} - 6 \sqrt{2}$

Decimal Form:

$1.05727969 \dots$