Calculus Examples

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

Step 1

Combine terms.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Reorder the factors of $x \cdot 3$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 2

Evaluate the limit.

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Step 2.1

Simplify the limit argument.

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Step 2.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.2

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

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

Reorder terms.

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

Step 2.1.3.5

Cancel the common factor.

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

Step 2.1.3.6

Rewrite the expression.

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

Step 2.2

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

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

Step 2.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

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

Step 2.4

Evaluate the limit of $- 1$ which is constant as $x$ approaches $3$ .

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

Step 3

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{1}{3} \cdot \frac{- 1}{3}$

Step 4

Simplify the answer.

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Step 4.1

Move the negative in front of the fraction.

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

Step 4.2

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

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Step 4.2.1

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

$- \frac{1}{3 \cdot 3}$

Step 4.2.2

Multiply $3$ by $3$ .

$- \frac{1}{9}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{9}$

Decimal Form:

$- 0.\overline{1}$