Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

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

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Step 16.1

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

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

Step 16.2

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

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

Step 16.3

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

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

Step 16.4

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

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

Step 17

Simplify the answer.

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Step 17.1

Simplify each term.

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Step 17.1.1

Simplify the numerator.

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Step 17.1.1.1

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

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

Step 17.1.1.2

Multiply $2$ by $64$ .

$\frac{128 + 1}{8 + 1} - \frac{2 \cdot 8^{2} + 1}{8 - 1 \cdot 1}$

Step 17.1.1.3

Add $128$ and $1$ .

$\frac{129}{8 + 1} - \frac{2 \cdot 8^{2} + 1}{8 - 1 \cdot 1}$

Step 17.1.2

Add $8$ and $1$ .

$\frac{129}{9} - \frac{2 \cdot 8^{2} + 1}{8 - 1 \cdot 1}$

Step 17.1.3

Cancel the common factor of $129$ and $9$ .

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Step 17.1.3.1

Factor $3$ out of $129$ .

$\frac{3 (43)}{9} - \frac{2 \cdot 8^{2} + 1}{8 - 1 \cdot 1}$

Step 17.1.3.2

Cancel the common factors.

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Step 17.1.3.2.1

Factor $3$ out of $9$ .

$\frac{3 \cdot 43}{3 \cdot 3} - \frac{2 \cdot 8^{2} + 1}{8 - 1 \cdot 1}$

Step 17.1.3.2.2

Cancel the common factor.

$\frac{3 \cdot 43}{3 \cdot 3} - \frac{2 \cdot 8^{2} + 1}{8 - 1 \cdot 1}$

Step 17.1.3.2.3

Rewrite the expression.

$\frac{43}{3} - \frac{2 \cdot 8^{2} + 1}{8 - 1 \cdot 1}$

Step 17.1.4

Simplify the numerator.

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Step 17.1.4.1

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

$\frac{43}{3} - \frac{2 \cdot 64 + 1}{8 - 1 \cdot 1}$

Step 17.1.4.2

Multiply $2$ by $64$ .

$\frac{43}{3} - \frac{128 + 1}{8 - 1 \cdot 1}$

Step 17.1.4.3

Add $128$ and $1$ .

$\frac{43}{3} - \frac{129}{8 - 1 \cdot 1}$

Step 17.1.5

Simplify the denominator.

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Step 17.1.5.1

Multiply $- 1$ by $1$ .

$\frac{43}{3} - \frac{129}{8 - 1}$

Step 17.1.5.2

Subtract $1$ from $8$ .

$\frac{43}{3} - \frac{129}{7}$

Step 17.2

To write $\frac{43}{3}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{43}{3} \cdot \frac{7}{7} - \frac{129}{7}$

Step 17.3

To write $- \frac{129}{7}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{43}{3} \cdot \frac{7}{7} - \frac{129}{7} \cdot \frac{3}{3}$

Step 17.4

Write each expression with a common denominator of $21$ , by multiplying each by an appropriate factor of $1$ .

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Step 17.4.1

Multiply $\frac{43}{3}$ by $\frac{7}{7}$ .

$\frac{43 \cdot 7}{3 \cdot 7} - \frac{129}{7} \cdot \frac{3}{3}$

Step 17.4.2

Multiply $3$ by $7$ .

$\frac{43 \cdot 7}{21} - \frac{129}{7} \cdot \frac{3}{3}$

Step 17.4.3

Multiply $\frac{129}{7}$ by $\frac{3}{3}$ .

$\frac{43 \cdot 7}{21} - \frac{129 \cdot 3}{7 \cdot 3}$

Step 17.4.4

Multiply $7$ by $3$ .

$\frac{43 \cdot 7}{21} - \frac{129 \cdot 3}{21}$

Step 17.5

Combine the numerators over the common denominator.

$\frac{43 \cdot 7 - 129 \cdot 3}{21}$

Step 17.6

Simplify the numerator.

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Step 17.6.1

Multiply $43$ by $7$ .

$\frac{301 - 129 \cdot 3}{21}$

Step 17.6.2

Multiply $- 129$ by $3$ .

$\frac{301 - 387}{21}$

Step 17.6.3

Subtract $387$ from $301$ .

$\frac{- 86}{21}$

Step 17.7

Move the negative in front of the fraction.

$- \frac{86}{21}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$- \frac{86}{21}$

Decimal Form:

$- 4.\overline{095238}$