Calculus Examples

$\frac{lim_{x \to 1} x^{2} - x}{x^{2} - 1}$

Step 1

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

$\frac{lim_{x \to 1} x^{2} - lim_{x \to 1} x}{x^{2} - 1}$

Step 2

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 1} x)}^{2} - lim_{x \to 1} x}{x^{2} - 1}$

Step 3

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 3.1

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

$\frac{1^{2} - lim_{x \to 1} x}{x^{2} - 1}$

Step 3.2

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

$\frac{1^{2} - 1}{x^{2} - 1}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

One to any power is one.

$\frac{1 - 1}{x^{2} - 1}$

Step 4.1.2

Subtract $1$ from $1$ .

$\frac{0}{x^{2} - 1}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{0}{x^{2} - 1^{2}}$

Step 4.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 4.3

Divide $0$ by $(x + 1) (x - 1)$ .

$0$