Calculus Examples

$lim_{n \to 8} \frac{3 n^{2} + 4 n + 5}{2 n^{2} - 3 n + 1}$

Step 1

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

$\frac{lim_{n \to 8} 3 n^{2} + 4 n + 5}{lim_{n \to 8} 2 n^{2} - 3 n + 1}$

Step 2

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

$\frac{lim_{n \to 8} 3 n^{2} + lim_{n \to 8} 4 n + lim_{n \to 8} 5}{lim_{n \to 8} 2 n^{2} - 3 n + 1}$

Step 3

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

$\frac{3 lim_{n \to 8} n^{2} + lim_{n \to 8} 4 n + lim_{n \to 8} 5}{lim_{n \to 8} 2 n^{2} - 3 n + 1}$

Step 4

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

$\frac{3 {(lim_{n \to 8} n)}^{2} + lim_{n \to 8} 4 n + lim_{n \to 8} 5}{lim_{n \to 8} 2 n^{2} - 3 n + 1}$

Step 5

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

$\frac{3 {(lim_{n \to 8} n)}^{2} + 4 lim_{n \to 8} n + lim_{n \to 8} 5}{lim_{n \to 8} 2 n^{2} - 3 n + 1}$

Step 6

Evaluate the limit of $5$ which is constant as $n$ approaches $8$ .

$\frac{3 {(lim_{n \to 8} n)}^{2} + 4 lim_{n \to 8} n + 5}{lim_{n \to 8} 2 n^{2} - 3 n + 1}$

Step 7

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

$\frac{3 {(lim_{n \to 8} n)}^{2} + 4 lim_{n \to 8} n + 5}{lim_{n \to 8} 2 n^{2} - lim_{n \to 8} 3 n + lim_{n \to 8} 1}$

Step 8

Move the term $2$ outside of the limit because it is constant with respect to $n$ .

$\frac{3 {(lim_{n \to 8} n)}^{2} + 4 lim_{n \to 8} n + 5}{2 lim_{n \to 8} n^{2} - lim_{n \to 8} 3 n + lim_{n \to 8} 1}$

Step 9

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

$\frac{3 {(lim_{n \to 8} n)}^{2} + 4 lim_{n \to 8} n + 5}{2 {(lim_{n \to 8} n)}^{2} - lim_{n \to 8} 3 n + lim_{n \to 8} 1}$

Step 10

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

$\frac{3 {(lim_{n \to 8} n)}^{2} + 4 lim_{n \to 8} n + 5}{2 {(lim_{n \to 8} n)}^{2} - 3 lim_{n \to 8} n + lim_{n \to 8} 1}$

Step 11

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

$\frac{3 {(lim_{n \to 8} n)}^{2} + 4 lim_{n \to 8} n + 5}{2 {(lim_{n \to 8} n)}^{2} - 3 lim_{n \to 8} n + 1}$

Step 12

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

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Step 12.1

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

$\frac{3 \cdot 8^{2} + 4 lim_{n \to 8} n + 5}{2 {(lim_{n \to 8} n)}^{2} - 3 lim_{n \to 8} n + 1}$

Step 12.2

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

$\frac{3 \cdot 8^{2} + 4 \cdot 8 + 5}{2 {(lim_{n \to 8} n)}^{2} - 3 lim_{n \to 8} n + 1}$

Step 12.3

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

$\frac{3 \cdot 8^{2} + 4 \cdot 8 + 5}{2 \cdot 8^{2} - 3 lim_{n \to 8} n + 1}$

Step 12.4

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

$\frac{3 \cdot 8^{2} + 4 \cdot 8 + 5}{2 \cdot 8^{2} - 3 \cdot 8 + 1}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

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

$\frac{3 \cdot 64 + 4 \cdot 8 + 5}{2 \cdot 8^{2} - 3 \cdot 8 + 1}$

Step 13.1.2

Multiply $3$ by $64$ .

$\frac{192 + 4 \cdot 8 + 5}{2 \cdot 8^{2} - 3 \cdot 8 + 1}$

Step 13.1.3

Multiply $4$ by $8$ .

$\frac{192 + 32 + 5}{2 \cdot 8^{2} - 3 \cdot 8 + 1}$

Step 13.1.4

Add $192$ and $32$ .

$\frac{224 + 5}{2 \cdot 8^{2} - 3 \cdot 8 + 1}$

Step 13.1.5

Add $224$ and $5$ .

$\frac{229}{2 \cdot 8^{2} - 3 \cdot 8 + 1}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

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

$\frac{229}{2 \cdot 64 - 3 \cdot 8 + 1}$

Step 13.2.2

Multiply $2$ by $64$ .

$\frac{229}{128 - 3 \cdot 8 + 1}$

Step 13.2.3

Multiply $- 3$ by $8$ .

$\frac{229}{128 - 24 + 1}$

Step 13.2.4

Subtract $24$ from $128$ .

$\frac{229}{104 + 1}$

Step 13.2.5

Add $104$ and $1$ .

$\frac{229}{105}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{229}{105}$

Decimal Form:

$2.18095238 \dots$