Calculus Examples

$lim_{x \to 4} \frac{5}{\frac{x}{x - 4}} - \frac{\frac{5}{4}}{x - 4}$

Step 1

Simplify the limit argument.

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Step 1.1

Simplify complex fractions.

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Step 1.1.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 4} 5 \frac{x - 4}{x} - \frac{\frac{5}{4}}{x - 4}$

Step 1.1.2

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 4} 5 \frac{x - 4}{x} - (\frac{5}{4} \cdot \frac{1}{x - 4})$

Step 1.2

Combine factors.

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Step 1.2.1

Combine $5$ and $\frac{x - 4}{x}$ .

$lim_{x \to 4} \frac{5 (x - 4)}{x} - (\frac{5}{4} \cdot \frac{1}{x - 4})$

Step 1.2.2

Multiply $\frac{5}{4}$ by $\frac{1}{x - 4}$ .

$lim_{x \to 4} \frac{5 (x - 4)}{x} - \frac{5}{4 (x - 4)}$

Step 1.3

Combine terms.

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Step 1.3.1

To write $\frac{5 (x - 4)}{x}$ as a fraction with a common denominator, multiply by $\frac{4 (x - 4)}{4 (x - 4)}$ .

$lim_{x \to 4} \frac{5 (x - 4)}{x} \cdot \frac{4 (x - 4)}{4 (x - 4)} - \frac{5}{4 (x - 4)}$

Step 1.3.2

To write $- \frac{5}{4 (x - 4)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$lim_{x \to 4} \frac{5 (x - 4)}{x} \cdot \frac{4 (x - 4)}{4 (x - 4)} - \frac{5}{4 (x - 4)} \cdot \frac{x}{x}$

Step 1.3.3

Write each expression with a common denominator of $x \cdot 4 (x - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.3.1

Multiply $\frac{5 (x - 4)}{x}$ by $\frac{4 (x - 4)}{4 (x - 4)}$ .

$lim_{x \to 4} \frac{5 (x - 4) (4 (x - 4))}{x (4 (x - 4))} - \frac{5}{4 (x - 4)} \cdot \frac{x}{x}$

Step 1.3.3.2

Multiply $\frac{5}{4 (x - 4)}$ by $\frac{x}{x}$ .

$lim_{x \to 4} \frac{5 (x - 4) (4 (x - 4))}{x (4 (x - 4))} - \frac{5 x}{4 (x - 4) x}$

Step 1.3.3.3

Reorder the factors of $x (4 (x - 4))$ .

$lim_{x \to 4} \frac{5 (x - 4) (4 (x - 4))}{4 x (x - 4)} - \frac{5 x}{4 (x - 4) x}$

Step 1.3.3.4

Reorder the factors of $4 (x - 4) x$ .

$lim_{x \to 4} \frac{5 (x - 4) (4 (x - 4))}{4 x (x - 4)} - \frac{5 x}{4 x (x - 4)}$

Step 1.3.4

Combine the numerators over the common denominator.

$lim_{x \to 4} \frac{5 (x - 4) (4 (x - 4)) - 5 x}{4 x (x - 4)}$

Step 2

Since the function approaches $\infty$ from the left and $- \infty$ from the right, the limit does not exist.

$Does not exist$