Calculus Examples

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

Step 1

Combine terms.

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Step 1.1

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

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

Step 1.2

Combine $- 1$ and $\frac{x}{x}$ .

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

Step 1.3

Combine the numerators over the common denominator.

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

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.

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

Step 2.1.2

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

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

Step 2.1.3

Cancel the common factor of $1 - x$ and $x - 1$ .

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

Reorder terms.

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

Step 2.1.3.5

Cancel the common factor.

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

Step 2.1.3.6

Rewrite the expression.

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

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

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

Step 2.3

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

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

Step 3

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{- 1}{1}$

Step 4

Divide $- 1$ by $1$ .

$- 1$