Calculus Examples

$lim_{n \to 8} \frac{3 n^{2} + 4 n + 1}{\sqrt{9 n^{4} + n^{3} + n + 1}}$

Step 1

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

$\frac{lim_{n \to 8} 3 n^{2} + 4 n + 1}{lim_{n \to 8} \sqrt{9 n^{4} + n^{3} + n + 1}}$

Step 2

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

$\frac{lim_{n \to 8} 3 n^{2} + lim_{n \to 8} 4 n + lim_{n \to 8} 1}{lim_{n \to 8} \sqrt{9 n^{4} + n^{3} + n + 1}}$

Step 3

Move the term $3$ outside of the limit because it is constant with respect to $n$ .

$\frac{3 lim_{n \to 8} n^{2} + lim_{n \to 8} 4 n + lim_{n \to 8} 1}{lim_{n \to 8} \sqrt{9 n^{4} + n^{3} + n + 1}}$

Step 4

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

$\frac{3 {(lim_{n \to 8} n)}^{2} + lim_{n \to 8} 4 n + lim_{n \to 8} 1}{lim_{n \to 8} \sqrt{9 n^{4} + n^{3} + n + 1}}$

Step 5

Move the term $4$ outside of the limit because it is constant with respect to $n$ .

$\frac{3 {(lim_{n \to 8} n)}^{2} + 4 lim_{n \to 8} n + lim_{n \to 8} 1}{lim_{n \to 8} \sqrt{9 n^{4} + n^{3} + n + 1}}$

Step 6

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

$\frac{3 {(lim_{n \to 8} n)}^{2} + 4 lim_{n \to 8} n + 1}{lim_{n \to 8} \sqrt{9 n^{4} + n^{3} + n + 1}}$

Step 7

Move the limit under the radical sign.

$\frac{3 {(lim_{n \to 8} n)}^{2} + 4 lim_{n \to 8} n + 1}{\sqrt{lim_{n \to 8} 9 n^{4} + n^{3} + n + 1}}$

Step 8

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

$\frac{3 {(lim_{n \to 8} n)}^{2} + 4 lim_{n \to 8} n + 1}{\sqrt{lim_{n \to 8} 9 n^{4} + lim_{n \to 8} n^{3} + lim_{n \to 8} n + lim_{n \to 8} 1}}$

Step 9

Move the term $9$ outside of the limit because it is constant with respect to $n$ .

$\frac{3 {(lim_{n \to 8} n)}^{2} + 4 lim_{n \to 8} n + 1}{\sqrt{9 lim_{n \to 8} n^{4} + lim_{n \to 8} n^{3} + lim_{n \to 8} n + lim_{n \to 8} 1}}$

Step 10

Move the exponent $4$ from $n^{4}$ outside the limit using the Limits Power Rule.

$\frac{3 {(lim_{n \to 8} n)}^{2} + 4 lim_{n \to 8} n + 1}{\sqrt{9 {(lim_{n \to 8} n)}^{4} + lim_{n \to 8} n^{3} + lim_{n \to 8} n + lim_{n \to 8} 1}}$

Step 11

Move the exponent $3$ from $n^{3}$ outside the limit using the Limits Power Rule.

$\frac{3 {(lim_{n \to 8} n)}^{2} + 4 lim_{n \to 8} n + 1}{\sqrt{9 {(lim_{n \to 8} n)}^{4} + {(lim_{n \to 8} n)}^{3} + lim_{n \to 8} n + lim_{n \to 8} 1}}$

Step 12

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

$\frac{3 {(lim_{n \to 8} n)}^{2} + 4 lim_{n \to 8} n + 1}{\sqrt{9 {(lim_{n \to 8} n)}^{4} + {(lim_{n \to 8} n)}^{3} + lim_{n \to 8} n + 1}}$

Step 13

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

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Step 13.1

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

$\frac{3 \cdot 8^{2} + 4 lim_{n \to 8} n + 1}{\sqrt{9 {(lim_{n \to 8} n)}^{4} + {(lim_{n \to 8} n)}^{3} + lim_{n \to 8} n + 1}}$

Step 13.2

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

$\frac{3 \cdot 8^{2} + 4 \cdot 8 + 1}{\sqrt{9 {(lim_{n \to 8} n)}^{4} + {(lim_{n \to 8} n)}^{3} + lim_{n \to 8} n + 1}}$

Step 13.3

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

$\frac{3 \cdot 8^{2} + 4 \cdot 8 + 1}{\sqrt{9 \cdot 8^{4} + {(lim_{n \to 8} n)}^{3} + lim_{n \to 8} n + 1}}$

Step 13.4

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

$\frac{3 \cdot 8^{2} + 4 \cdot 8 + 1}{\sqrt{9 \cdot 8^{4} + 8^{3} + lim_{n \to 8} n + 1}}$

Step 13.5

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

$\frac{3 \cdot 8^{2} + 4 \cdot 8 + 1}{\sqrt{9 \cdot 8^{4} + 8^{3} + 8 + 1}}$

$\frac{3 \cdot 8^{2} + 4 \cdot 8 + 1}{\sqrt{9 \cdot 8^{4} + 8^{3} + 8 + 1}}$

Step 14

Simplify the answer.

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Step 14.1

Simplify the numerator.

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Step 14.1.1

Raise $8$ to the power of $2$ .

$\frac{3 \cdot 64 + 4 \cdot 8 + 1}{\sqrt{9 \cdot 8^{4} + 8^{3} + 8 + 1}}$

Step 14.1.2

Multiply $3$ by $64$ .

$\frac{192 + 4 \cdot 8 + 1}{\sqrt{9 \cdot 8^{4} + 8^{3} + 8 + 1}}$

Step 14.1.3

Multiply $4$ by $8$ .

$\frac{192 + 32 + 1}{\sqrt{9 \cdot 8^{4} + 8^{3} + 8 + 1}}$

Step 14.1.4

Add $192$ and $32$ .

$\frac{224 + 1}{\sqrt{9 \cdot 8^{4} + 8^{3} + 8 + 1}}$

Step 14.1.5

Add $224$ and $1$ .

$\frac{225}{\sqrt{9 \cdot 8^{4} + 8^{3} + 8 + 1}}$

$\frac{225}{\sqrt{9 \cdot 8^{4} + 8^{3} + 8 + 1}}$

Step 14.2

Simplify the denominator.

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Step 14.2.1

Raise $8$ to the power of $4$ .

$\frac{225}{\sqrt{9 \cdot 4096 + 8^{3} + 8 + 1}}$

Step 14.2.2

Multiply $9$ by $4096$ .

$\frac{225}{\sqrt{36864 + 8^{3} + 8 + 1}}$

Step 14.2.3

Raise $8$ to the power of $3$ .

$\frac{225}{\sqrt{36864 + 512 + 8 + 1}}$

Step 14.2.4

Add $36864$ and $512$ .

$\frac{225}{\sqrt{37376 + 8 + 1}}$

Step 14.2.5

Add $37376$ and $8$ .

$\frac{225}{\sqrt{37384 + 1}}$

Step 14.2.6

Add $37384$ and $1$ .

$\frac{225}{\sqrt{37385}}$

$\frac{225}{\sqrt{37385}}$

Step 14.3

Multiply $\frac{225}{\sqrt{37385}}$ by $\frac{\sqrt{37385}}{\sqrt{37385}}$ .

$\frac{225}{\sqrt{37385}} \cdot \frac{\sqrt{37385}}{\sqrt{37385}}$

Step 14.4

Combine and simplify the denominator.

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Step 14.4.1

Multiply $\frac{225}{\sqrt{37385}}$ by $\frac{\sqrt{37385}}{\sqrt{37385}}$ .

$\frac{225 \sqrt{37385}}{\sqrt{37385} \sqrt{37385}}$

Step 14.4.2

Raise $\sqrt{37385}$ to the power of $1$ .

$\frac{225 \sqrt{37385}}{{\sqrt{37385}}^{1} \sqrt{37385}}$

Step 14.4.3

Raise $\sqrt{37385}$ to the power of $1$ .

$\frac{225 \sqrt{37385}}{{\sqrt{37385}}^{1} {\sqrt{37385}}^{1}}$

Step 14.4.4

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

$\frac{225 \sqrt{37385}}{{\sqrt{37385}}^{1 + 1}}$

Step 14.4.5

Add $1$ and $1$ .

$\frac{225 \sqrt{37385}}{{\sqrt{37385}}^{2}}$

Step 14.4.6

Rewrite ${\sqrt{37385}}^{2}$ as $37385$ .

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Step 14.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{37385}$ as $37385^{\frac{1}{2}}$ .

$\frac{225 \sqrt{37385}}{{(37385^{\frac{1}{2}})}^{2}}$

Step 14.4.6.2

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

$\frac{225 \sqrt{37385}}{37385^{\frac{1}{2} \cdot 2}}$

Step 14.4.6.3

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

$\frac{225 \sqrt{37385}}{37385^{\frac{2}{2}}}$

Step 14.4.6.4

Cancel the common factor of $2$ .

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Step 14.4.6.4.1

Cancel the common factor.

$\frac{225 \sqrt{37385}}{37385^{\frac{2}{2}}}$

Step 14.4.6.4.2

Rewrite the expression.

$\frac{225 \sqrt{37385}}{37385^{1}}$

$\frac{225 \sqrt{37385}}{37385^{1}}$

Step 14.4.6.5

Evaluate the exponent.

$\frac{225 \sqrt{37385}}{37385}$

$\frac{225 \sqrt{37385}}{37385}$

$\frac{225 \sqrt{37385}}{37385}$

Step 14.5

Cancel the common factor of $225$ and $37385$ .

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Step 14.5.1

Factor $5$ out of $225 \sqrt{37385}$ .

$\frac{5 (45 \sqrt{37385})}{37385}$

Step 14.5.2

Cancel the common factors.

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Step 14.5.2.1

Factor $5$ out of $37385$ .

$\frac{5 (45 \sqrt{37385})}{5 (7477)}$

Step 14.5.2.2

Cancel the common factor.

$\frac{5 (45 \sqrt{37385})}{5 \cdot 7477}$

Step 14.5.2.3

Rewrite the expression.

$\frac{45 \sqrt{37385}}{7477}$

$\frac{45 \sqrt{37385}}{7477}$

$\frac{45 \sqrt{37385}}{7477}$

$\frac{45 \sqrt{37385}}{7477}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{45 \sqrt{37385}}{7477}$

Decimal Form:

$1.16368068 \dots$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$