Calculus Examples

$lim_{x \to 0} \frac{\frac{1}{x} - \frac{1}{3}}{x - 3}$

Step 1

Combine terms.

Tap for more steps...

Step 1.1

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$lim_{x \to 0} \frac{\frac{1}{x} \cdot \frac{3}{3} - \frac{1}{3}}{x - 3}$

Step 1.2

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$lim_{x \to 0} \frac{\frac{1}{x} \cdot \frac{3}{3} - \frac{1}{3} \cdot \frac{x}{x}}{x - 3}$

Step 1.3

Write each expression with a common denominator of $x \cdot 3$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.3.1

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

$lim_{x \to 0} \frac{\frac{3}{x \cdot 3} - \frac{1}{3} \cdot \frac{x}{x}}{x - 3}$

Step 1.3.2

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

$lim_{x \to 0} \frac{\frac{3}{x \cdot 3} - \frac{x}{3 x}}{x - 3}$

Step 1.3.3

Reorder the factors of $x \cdot 3$ .

$lim_{x \to 0} \frac{\frac{3}{3 x} - \frac{x}{3 x}}{x - 3}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{3 - x}{3 x}}{x - 3}$

Step 2

Simplify the limit argument.

Tap for more steps...

Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \frac{3 - x}{3 x} \cdot \frac{1}{x - 3}$

Step 2.2

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

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

Step 2.3

Cancel the common factor of $3 - x$ and $x - 3$ .

Tap for more steps...

Step 2.3.1

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 2.3.2

Factor $- 1$ out of $- x$ .

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

Step 2.3.3

Factor $- 1$ out of $- 1 (- 3) - (x)$ .

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

Step 2.3.4

Reorder terms.

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

Step 2.3.5

Cancel the common factor.

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

Step 2.3.6

Rewrite the expression.

$lim_{x \to 0} \frac{- 1}{3 x}$

Step 3

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

$Does not exist$