Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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Step 13.1

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

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

Step 13.2

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

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

Step 13.3

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

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

Step 13.4

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

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

Step 14

Simplify the answer.

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Step 14.1

Simplify the numerator.

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Step 14.1.1

Multiply $2$ by $8$ .

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

Step 14.1.2

Multiply $- 1$ by $1$ .

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

Step 14.1.3

Subtract $1$ from $16$ .

$\frac{15 (3 - 8)}{(8 - 1 \cdot 1) \cdot (8 + 3)}$

Step 14.1.4

Subtract $8$ from $3$ .

$\frac{15 \cdot - 5}{(8 - 1 \cdot 1) \cdot (8 + 3)}$

Step 14.2

Simplify the denominator.

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Step 14.2.1

Multiply $- 1$ by $1$ .

$\frac{15 \cdot - 5}{(8 - 1) (8 + 3)}$

Step 14.2.2

Subtract $1$ from $8$ .

$\frac{15 \cdot - 5}{7 (8 + 3)}$

Step 14.2.3

Add $8$ and $3$ .

$\frac{15 \cdot - 5}{7 \cdot 11}$

Step 14.3

Multiply $15$ by $- 5$ .

$\frac{- 75}{7 \cdot 11}$

Step 14.4

Multiply $7$ by $11$ .

$\frac{- 75}{77}$

Step 14.5

Move the negative in front of the fraction.

$- \frac{75}{77}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$- \frac{75}{77}$

Decimal Form:

$- 0.\overline{974025}$