Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Multiply $2$ by $8$ .

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

Step 8.1.2

Multiply $- 1$ by $1$ .

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

Step 8.1.3

Subtract $1$ from $16$ .

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

Step 8.1.4

Subtract $8$ from $3$ .

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

Step 8.2

Multiply $15$ by $- 5$ .

$\frac{- 75}{(x - 1) (x + 3)}$

Step 8.3

Move the negative in front of the fraction.

$- \frac{75}{(x - 1) (x + 3)}$