Calculus Examples

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

Step 1

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

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

Step 2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $8$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

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

Step 9

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $8$ .

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

Step 10

Move the exponent $2$ from ${(2 x + 1)}^{2}$ outside the limit using the Limits Power Rule.

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

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

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Step 16.1

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

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

Step 16.2

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

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

Step 16.3

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

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

Step 16.4

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

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

Step 17

Simplify the answer.

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Step 17.1

Simplify the numerator.

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Step 17.1.1

Multiply $3$ by $8$ .

$\frac{(24 - 1 \cdot 1) (8^{2} - 1 \cdot 4)}{{(2 \cdot 8 + 1)}^{2} \cdot (8 - 1 \cdot 1)}$

Step 17.1.2

Multiply $- 1$ by $1$ .

$\frac{(24 - 1) (8^{2} - 1 \cdot 4)}{{(2 \cdot 8 + 1)}^{2} \cdot (8 - 1 \cdot 1)}$

Step 17.1.3

Subtract $1$ from $24$ .

$\frac{23 (8^{2} - 1 \cdot 4)}{{(2 \cdot 8 + 1)}^{2} \cdot (8 - 1 \cdot 1)}$

Step 17.1.4

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

$\frac{23 (64 - 1 \cdot 4)}{{(2 \cdot 8 + 1)}^{2} \cdot (8 - 1 \cdot 1)}$

Step 17.1.5

Multiply $- 1$ by $4$ .

$\frac{23 (64 - 4)}{{(2 \cdot 8 + 1)}^{2} \cdot (8 - 1 \cdot 1)}$

Step 17.1.6

Subtract $4$ from $64$ .

$\frac{23 \cdot 60}{{(2 \cdot 8 + 1)}^{2} \cdot (8 - 1 \cdot 1)}$

Step 17.2

Simplify the denominator.

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Step 17.2.1

Multiply $2$ by $8$ .

$\frac{23 \cdot 60}{{(16 + 1)}^{2} (8 - 1 \cdot 1)}$

Step 17.2.2

Add $16$ and $1$ .

$\frac{23 \cdot 60}{17^{2} (8 - 1 \cdot 1)}$

Step 17.2.3

Multiply $- 1$ by $1$ .

$\frac{23 \cdot 60}{17^{2} (8 - 1)}$

Step 17.2.4

Subtract $1$ from $8$ .

$\frac{23 \cdot 60}{17^{2} \cdot 7}$

Step 17.2.5

Raise $17$ to the power of $2$ .

$\frac{23 \cdot 60}{289 \cdot 7}$

Step 17.3

Multiply $23$ by $60$ .

$\frac{1380}{289 \cdot 7}$

Step 17.4

Multiply $289$ by $7$ .

$\frac{1380}{2023}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$\frac{1380}{2023}$

Decimal Form:

$0.68215521 \dots$