Calculus Examples

$lim_{x \to 8} \frac{(6 x - 2) (2 x - 3)}{(3 x + 2) (4 x - 1) (x - 1)}$

Step 1

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

$\frac{lim_{x \to 8} (6 x - 2) (2 x - 3)}{lim_{x \to 8} (3 x + 2) (4 x - 1) (x - 1)}$

Step 2

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

$\frac{lim_{x \to 8} 6 x - 2 \cdot lim_{x \to 8} 2 x - 3}{lim_{x \to 8} (3 x + 2) (4 x - 1) (x - 1)}$

Step 3

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

$\frac{(lim_{x \to 8} 6 x - lim_{x \to 8} 2) \cdot lim_{x \to 8} 2 x - 3}{lim_{x \to 8} (3 x + 2) (4 x - 1) (x - 1)}$

Step 4

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{(6 lim_{x \to 8} x - lim_{x \to 8} 2) \cdot lim_{x \to 8} 2 x - 3}{lim_{x \to 8} (3 x + 2) (4 x - 1) (x - 1)}$

Step 5

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

$\frac{(6 lim_{x \to 8} x - 1 \cdot 2) \cdot lim_{x \to 8} 2 x - 3}{lim_{x \to 8} (3 x + 2) (4 x - 1) (x - 1)}$

Step 6

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

$\frac{(6 lim_{x \to 8} x - 1 \cdot 2) \cdot (lim_{x \to 8} 2 x - lim_{x \to 8} 3)}{lim_{x \to 8} (3 x + 2) (4 x - 1) (x - 1)}$

Step 7

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

$\frac{(6 lim_{x \to 8} x - 1 \cdot 2) \cdot (2 lim_{x \to 8} x - lim_{x \to 8} 3)}{lim_{x \to 8} (3 x + 2) (4 x - 1) (x - 1)}$

Step 8

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

$\frac{(6 lim_{x \to 8} x - 1 \cdot 2) \cdot (2 lim_{x \to 8} x - 1 \cdot 3)}{lim_{x \to 8} (3 x + 2) (4 x - 1) (x - 1)}$

Step 9

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

$\frac{(6 lim_{x \to 8} x - 1 \cdot 2) \cdot (2 lim_{x \to 8} x - 1 \cdot 3)}{lim_{x \to 8} 3 x + 2 \cdot lim_{x \to 8} 4 x - 1 \cdot lim_{x \to 8} x - 1}$

Step 10

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

$\frac{(6 lim_{x \to 8} x - 1 \cdot 2) \cdot (2 lim_{x \to 8} x - 1 \cdot 3)}{(lim_{x \to 8} 3 x + lim_{x \to 8} 2) \cdot lim_{x \to 8} 4 x - 1 \cdot lim_{x \to 8} x - 1}$

Step 11

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

$\frac{(6 lim_{x \to 8} x - 1 \cdot 2) \cdot (2 lim_{x \to 8} x - 1 \cdot 3)}{(3 lim_{x \to 8} x + lim_{x \to 8} 2) \cdot lim_{x \to 8} 4 x - 1 \cdot lim_{x \to 8} x - 1}$

Step 12

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

$\frac{(6 lim_{x \to 8} x - 1 \cdot 2) \cdot (2 lim_{x \to 8} x - 1 \cdot 3)}{(3 lim_{x \to 8} x + 2) \cdot lim_{x \to 8} 4 x - 1 \cdot lim_{x \to 8} x - 1}$

Step 13

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

$\frac{(6 lim_{x \to 8} x - 1 \cdot 2) \cdot (2 lim_{x \to 8} x - 1 \cdot 3)}{(3 lim_{x \to 8} x + 2) \cdot (lim_{x \to 8} 4 x - lim_{x \to 8} 1) \cdot lim_{x \to 8} x - 1}$

Step 14

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

$\frac{(6 lim_{x \to 8} x - 1 \cdot 2) \cdot (2 lim_{x \to 8} x - 1 \cdot 3)}{(3 lim_{x \to 8} x + 2) \cdot (4 lim_{x \to 8} x - lim_{x \to 8} 1) \cdot lim_{x \to 8} x - 1}$

Step 15

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

$\frac{(6 lim_{x \to 8} x - 1 \cdot 2) \cdot (2 lim_{x \to 8} x - 1 \cdot 3)}{(3 lim_{x \to 8} x + 2) \cdot (4 lim_{x \to 8} x - 1 \cdot 1) \cdot lim_{x \to 8} x - 1}$

Step 16

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

$\frac{(6 lim_{x \to 8} x - 1 \cdot 2) \cdot (2 lim_{x \to 8} x - 1 \cdot 3)}{(3 lim_{x \to 8} x + 2) \cdot (4 lim_{x \to 8} x - 1 \cdot 1) \cdot (lim_{x \to 8} x - lim_{x \to 8} 1)}$

Step 17

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

$\frac{(6 lim_{x \to 8} x - 1 \cdot 2) \cdot (2 lim_{x \to 8} x - 1 \cdot 3)}{(3 lim_{x \to 8} x + 2) \cdot (4 lim_{x \to 8} x - 1 \cdot 1) \cdot (lim_{x \to 8} x - 1 \cdot 1)}$

Step 18

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

Tap for more steps...

Step 18.1