Calculus Examples

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

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

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as

x

$x$ approaches

1

$1$ .

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

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

Step 1.2

Move the exponent

2

$2$ from

x^{2}

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

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

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

Step 1.3

Evaluate the limit of

1

$1$ which is constant as

x

$x$ approaches

1

$1$ .

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

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

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

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

Step 2

Evaluate the limit of

x

$x$ by plugging in

1

$1$ for

x

$x$ .

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

One to any power is one.

\frac{1 - 1 \cdot 1}{x - 1}

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

Step 3.1.2

Multiply

- 1

$- 1$ by

1

$1$ .

\frac{1 - 1}{x - 1}

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

Step 3.1.3

Subtract

1

$1$ from

1

$1$ .

\frac{0}{x - 1}

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

\frac{0}{x - 1}

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

Step 3.2

Divide

0

$0$ by

x - 1

$x - 1$ .

0

$0$

0

$0$

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

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

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