Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

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 $x$ .

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

Step 15.2

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

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

Step 15.3

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

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

Step 15.4

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

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

Step 16

Simplify the answer.

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Step 16.1

Simplify the numerator.

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Step 16.1.1

Multiply $3$ by $8$ .

$\frac{(24 + 1) (2 \cdot 8 - 1 \cdot 3)}{(5 \cdot 8 - 1 \cdot 2) \cdot (8 + 4)}$

Step 16.1.2

Add $24$ and $1$ .

$\frac{25 (2 \cdot 8 - 1 \cdot 3)}{(5 \cdot 8 - 1 \cdot 2) \cdot (8 + 4)}$

Step 16.1.3

Multiply $2$ by $8$ .

$\frac{25 (16 - 1 \cdot 3)}{(5 \cdot 8 - 1 \cdot 2) \cdot (8 + 4)}$

Step 16.1.4

Multiply $- 1$ by $3$ .

$\frac{25 (16 - 3)}{(5 \cdot 8 - 1 \cdot 2) \cdot (8 + 4)}$

Step 16.1.5

Subtract $3$ from $16$ .

$\frac{25 \cdot 13}{(5 \cdot 8 - 1 \cdot 2) \cdot (8 + 4)}$

Step 16.2

Simplify the denominator.

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Step 16.2.1

Multiply $5$ by $8$ .

$\frac{25 \cdot 13}{(40 - 1 \cdot 2) (8 + 4)}$

Step 16.2.2

Multiply $- 1$ by $2$ .

$\frac{25 \cdot 13}{(40 - 2) (8 + 4)}$

Step 16.2.3

Subtract $2$ from $40$ .

$\frac{25 \cdot 13}{38 (8 + 4)}$

Step 16.2.4

Add $8$ and $4$ .

$\frac{25 \cdot 13}{38 \cdot 12}$

Step 16.3

Multiply $25$ by $13$ .

$\frac{325}{38 \cdot 12}$

Step 16.4

Multiply $38$ by $12$ .

$\frac{325}{456}$

Step 17

The result can be shown in multiple forms.

Exact Form:

$\frac{325}{456}$

Decimal Form:

$0.71271929 \dots$