Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Move the exponent $2 - 1$ from $x^{2 - 1}$ outside the limit using the Limits Power Rule.

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $- 1$ by $1$ .

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

Step 6.1.2

Subtract $1$ from $1$ .

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

Step 6.2

Simplify the denominator.

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Step 6.2.1

Subtract $1$ from $2$ .

$\frac{0}{1^{1}}$

Step 6.2.2

Evaluate the exponent.

$\frac{0}{1}$

Step 6.3

Divide $0$ by $1$ .

$0$