Calculus Examples

$\frac{lim_{x \to 8} 7 x - 56}{8 x - 64}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{x \to 8} 7 x - lim_{x \to 8} 56}{8 x - 64}$

Step 1.2

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{7 lim_{x \to 8} x - lim_{x \to 8} 56}{8 x - 64}$

Step 1.3

Evaluate the limit of $56$ which is constant as $x$ approaches $8$ .

$\frac{7 lim_{x \to 8} x - 1 \cdot 56}{8 x - 64}$

Step 2

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

$\frac{7 \cdot 8 - 1 \cdot 56}{8 x - 64}$

Step 3

Simplify the answer.

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Step 3.1

Cancel the common factor of $7 \cdot 8 - 1 \cdot 56$ and $8 x - 64$ .

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Step 3.1.1

Factor $- 1$ out of $7 \cdot 8$ .

$\frac{- (- 7 \cdot 8) - 1 \cdot 56}{8 x - 64}$

Step 3.1.2

Factor $- 1$ out of $- 1 \cdot 56$ .

$\frac{- (- 7 \cdot 8) - 1 (56)}{8 x - 64}$

Step 3.1.3

Factor $- 1$ out of $- (- 7 \cdot 8) - 1 (56)$ .

$\frac{- (- 7 \cdot 8 + 56)}{8 x - 64}$

Step 3.1.4

Rewrite $- (- 7 \cdot 8 + 56)$ as $- 1 (- 7 \cdot 8 + 56)$ .

$\frac{- 1 (- 7 \cdot 8 + 56)}{8 x - 64}$

Step 3.1.5

Factor $8$ out of $- 1 (- 7 \cdot 8 + 56)$ .

$\frac{8 (- 1 (- 7 + 7))}{8 x - 64}$

Step 3.1.6

Cancel the common factors.

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Step 3.1.6.1

Factor $8$ out of $8 x$ .

$\frac{8 (- 1 (- 7 + 7))}{8 (x) - 64}$

Step 3.1.6.2

Factor $8$ out of $- 64$ .

$\frac{8 (- 1 (- 7 + 7))}{8 x + 8 \cdot - 8}$

Step 3.1.6.3

Factor $8$ out of $8 x + 8 \cdot - 8$ .

$\frac{8 (- 1 (- 7 + 7))}{8 (x - 8)}$

Step 3.1.6.4

Cancel the common factor.

$\frac{8 (- 1 (- 7 + 7))}{8 (x - 8)}$

Step 3.1.6.5

Rewrite the expression.

$\frac{- 1 (- 7 + 7)}{x - 8}$

Step 3.2

Add $- 7$ and $7$ .

$\frac{- 1 \cdot 0}{x - 8}$

Step 3.3

Multiply $- 1$ by $0$ .

$\frac{0}{x - 8}$

Step 3.4

Divide $0$ by $x - 8$ .

$0$