Calculus Examples

$lim_{x \to 8} \frac{7 x}{4 x - 8}$

Step 1

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

$7 lim_{x \to 8} \frac{x}{4 x - 8}$

Step 2

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

$7 \frac{lim_{x \to 8} x}{lim_{x \to 8} 4 x - 8}$

Step 3

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

$7 \frac{lim_{x \to 8} x}{lim_{x \to 8} 4 x - lim_{x \to 8} 8}$

Step 4

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

$7 \frac{lim_{x \to 8} x}{4 lim_{x \to 8} x - lim_{x \to 8} 8}$

Step 5

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

$7 \frac{lim_{x \to 8} x}{4 lim_{x \to 8} x - 1 \cdot 8}$

Step 6

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

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Step 6.1

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

$7 \frac{8}{4 lim_{x \to 8} x - 1 \cdot 8}$

Step 6.2

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

$7 \frac{8}{4 \cdot 8 - 1 \cdot 8}$

Step 7

Simplify the answer.

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Step 7.1

Cancel the common factor of $8$ and $4 \cdot 8 - 1 \cdot 8$ .

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Step 7.1.1

Factor $- 1$ out of $4 \cdot 8$ .

$7 \frac{8}{- (- 4 \cdot 8) - 1 \cdot 8}$

Step 7.1.2

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

$7 \frac{8}{- (- 4 \cdot 8) - 1 (8)}$

Step 7.1.3

Factor $- 1$ out of $- (- 4 \cdot 8) - 1 (8)$ .

$7 \frac{8}{- (- 4 \cdot 8 + 8)}$

Step 7.1.4

Rewrite $- (- 4 \cdot 8 + 8)$ as $- 1 (- 4 \cdot 8 + 8)$ .

$7 \frac{8}{- 1 (- 4 \cdot 8 + 8)}$

Step 7.1.5

Factor $8$ out of $8$ .

$7 \frac{8 (1)}{- 1 (- 4 \cdot 8 + 8)}$

Step 7.1.6

Cancel the common factors.

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Step 7.1.6.1

Factor $8$ out of $- 1 (- 4 \cdot 8 + 8)$ .

$7 \frac{8 (1)}{8 (- 1 (- 4 + 1))}$

Step 7.1.6.2

Cancel the common factor.

$7 \frac{8 \cdot 1}{8 (- 1 (- 4 + 1))}$

Step 7.1.6.3

Rewrite the expression.

$7 \frac{1}{- 1 (- 4 + 1)}$

Step 7.2

Cancel the common factor of $1$ and $- 1$ .

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Step 7.2.1

Rewrite $1$ as $- 1 (- 1)$ .

$7 \frac{- 1 (- 1)}{- 1 (- 4 + 1)}$

Step 7.2.2

Move the negative in front of the fraction.

$7 (- \frac{1}{- 4 + 1})$

Step 7.3

Add $- 4$ and $1$ .

$7 (- \frac{1}{- 3})$

Step 7.4

Move the negative in front of the fraction.

$7 (- - \frac{1}{3})$

Step 7.5

Multiply $- - \frac{1}{3}$ .

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Step 7.5.1

Multiply $- 1$ by $- 1$ .

$7 (1 (\frac{1}{3}))$

Step 7.5.2

Multiply $\frac{1}{3}$ by $1$ .

$7 (\frac{1}{3})$

Step 7.6

Combine $7$ and $\frac{1}{3}$ .

$\frac{7}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{3}$

Decimal Form:

$2.\overline{3}$