Calculus Examples

$lim_{x \to 0^{+}} \frac{5}{x} - \frac{2}{3 x^{2}}$

Step 1

Combine terms.

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Step 1.1

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

$lim_{x \to 0^{+}} \frac{5}{x} \cdot \frac{3 x}{3 x} - \frac{2}{3 x^{2}}$

Step 1.2

Write each expression with a common denominator of $3 x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.1

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

$lim_{x \to 0^{+}} \frac{5 (3 x)}{x (3 x)} - \frac{2}{3 x^{2}}$

Step 1.2.2

Rewrite using the commutative property of multiplication.

$lim_{x \to 0^{+}} \frac{5 (3 x)}{3 x \cdot x} - \frac{2}{3 x^{2}}$

Step 1.2.3

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.3.1

Move $x$ .

$lim_{x \to 0^{+}} \frac{5 (3 x)}{3 (x \cdot x)} - \frac{2}{3 x^{2}}$

Step 1.2.3.2

Multiply $x$ by $x$ .

$lim_{x \to 0^{+}} \frac{5 (3 x)}{3 x^{2}} - \frac{2}{3 x^{2}}$

Step 1.3

Combine the numerators over the common denominator.

$lim_{x \to 0^{+}} \frac{5 (3 x) - 2}{3 x^{2}}$

Step 2

Since the numerator is negative and the denominator $3 x^{2}$ approaches zero and is greater than zero for $x$ near $0$ to the right, the function decreases without bound.

$- \infty$