Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Evaluate the limit of $3$ which is constant as $x$ approaches $8$ .

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

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

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Step 15.1

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

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

Step 15.2

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

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

Step 15.3

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

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

Step 15.4

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

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

Step 16

Simplify the answer.

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Step 16.1

Cancel the common factor of $8 - 1 \cdot 2$ and $3 \cdot 8 + 2$ .

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Step 16.1.1

Rewrite $8$ as $- 1 (- 8)$ .

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

Step 16.1.2

Factor $- 1$ out of $- 1 \cdot 2$ .

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

Step 16.1.3

Factor $- 1$ out of $- 1 (- 8) - 1 (2)$ .

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

Step 16.1.4

Factor $2$ out of $- 1 (- 8 + 2) \cdot (2 \cdot 8 - 1 \cdot 3)$ .

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

Step 16.1.5

Cancel the common factors.

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Step 16.1.5.1

Factor $2$ out of $(3 \cdot 8 + 2) \cdot (4 \cdot 8 - 1 \cdot 1)$ .

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

Step 16.1.5.2

Cancel the common factor.

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

Step 16.1.5.3

Rewrite the expression.

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

Step 16.2

Simplify the numerator.

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Step 16.2.1

Combine exponents.

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Step 16.2.1.1

Multiply $2$ by $8$ .

$\frac{- 1 (- 4 + 1) (16 - 1 \cdot 3)}{(3 \cdot 4 + 1) \cdot (4 \cdot 8 - 1 \cdot 1)}$

Step 16.2.1.2

Multiply $- 1$ by $3$ .

$\frac{- 1 (- 4 + 1) (16 - 3)}{(3 \cdot 4 + 1) \cdot (4 \cdot 8 - 1 \cdot 1)}$

Step 16.2.2

Add $- 4$ and $1$ .

$\frac{- 1 \cdot - 3 (16 - 3)}{(3 \cdot 4 + 1) \cdot (4 \cdot 8 - 1 \cdot 1)}$

Step 16.2.3

Subtract $3$ from $16$ .

$\frac{- 1 \cdot - 3 \cdot 13}{(3 \cdot 4 + 1) \cdot (4 \cdot 8 - 1 \cdot 1)}$

Step 16.2.4

Combine exponents.

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Step 16.2.4.1

Multiply $- 1$ by $- 3$ .

$\frac{3 \cdot 13}{(3 \cdot 4 + 1) \cdot (4 \cdot 8 - 1 \cdot 1)}$

Step 16.2.4.2

Multiply $3$ by $13$ .

$\frac{39}{(3 \cdot 4 + 1) \cdot (4 \cdot 8 - 1 \cdot 1)}$

Step 16.3

Simplify the denominator.

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Step 16.3.1

Multiply $3$ by $4$ .

$\frac{39}{(12 + 1) (4 \cdot 8 - 1 \cdot 1)}$

Step 16.3.2

Add $12$ and $1$ .

$\frac{39}{13 (4 \cdot 8 - 1 \cdot 1)}$

Step 16.3.3

Multiply $4$ by $8$ .

$\frac{39}{13 (32 - 1 \cdot 1)}$

Step 16.3.4

Multiply $- 1$ by $1$ .

$\frac{39}{13 (32 - 1)}$

Step 16.3.5

Subtract $1$ from $32$ .

$\frac{39}{13 \cdot 31}$

Step 16.4

Multiply $13$ by $31$ .

$\frac{39}{403}$

Step 16.5

Cancel the common factor of $39$ and $403$ .

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Step 16.5.1

Factor $13$ out of $39$ .

$\frac{13 (3)}{403}$

Step 16.5.2

Cancel the common factors.

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Step 16.5.2.1

Factor $13$ out of $403$ .

$\frac{13 \cdot 3}{13 \cdot 31}$

Step 16.5.2.2

Cancel the common factor.

$\frac{13 \cdot 3}{13 \cdot 31}$

Step 16.5.2.3

Rewrite the expression.

$\frac{3}{31}$

Step 17

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{31}$

Decimal Form:

$0.09677419 \dots$