Calculus Examples

$lim_{x \to 0} \frac{3}{x} \cdot (\frac{1}{5 + x}) - (\frac{1}{5 - x})$

Step 1

Simplify the limit argument.

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Step 1.1

Multiply $\frac{3}{x}$ by $\frac{1}{5 + x}$ .

$lim_{x \to 0} \frac{3}{x (5 + x)} - (\frac{1}{5 - x})$

Step 1.2

Combine terms.

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Step 1.2.1

To write $\frac{3}{x (5 + x)}$ as a fraction with a common denominator, multiply by $\frac{5 - x}{5 - x}$ .

$lim_{x \to 0} \frac{3}{x (5 + x)} \cdot \frac{5 - x}{5 - x} - (\frac{1}{5 - x})$

Step 1.2.2

To write $- (\frac{1}{5 - x})$ as a fraction with a common denominator, multiply by $\frac{x (5 + x)}{x (5 + x)}$ .

$lim_{x \to 0} \frac{3}{x (5 + x)} \cdot \frac{5 - x}{5 - x} - (\frac{1}{5 - x}) \cdot \frac{x (5 + x)}{x (5 + x)}$

Step 1.2.3

Write each expression with a common denominator of $x (5 + x) (5 - x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.3.1

Multiply $\frac{3}{x (5 + x)}$ by $\frac{5 - x}{5 - x}$ .

$lim_{x \to 0} \frac{3 (5 - x)}{x (5 + x) (5 - x)} - (\frac{1}{5 - x}) \cdot \frac{x (5 + x)}{x (5 + x)}$

Step 1.2.3.2

Multiply $\frac{1}{5 - x}$ by $\frac{x (5 + x)}{x (5 + x)}$ .

$lim_{x \to 0} \frac{3 (5 - x)}{x (5 + x) (5 - x)} - \frac{x (5 + x)}{(5 - x) (x (5 + x))}$

Step 1.2.3.3

Reorder the factors of $(5 - x) (x (5 + x))$ .

$lim_{x \to 0} \frac{3 (5 - x)}{x (5 + x) (5 - x)} - \frac{x (5 + x)}{x (5 + x) (5 - x)}$

Step 1.2.4

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{3 (5 - x) - x (5 + x)}{x (5 + x) (5 - x)}$

Step 2

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

$Does not exist$