Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Evaluate the limit of $2$ which is constant as $x$ approaches $1$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

One to any power is one.

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

Step 9.1.2

Multiply $- 1$ by $2$ .

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

Step 9.1.3

Subtract $1$ from $1$ .

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

Step 9.1.4

Subtract $2$ from $0$ .

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

Step 9.2

Simplify the denominator.

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Step 9.2.1

One to any power is one.

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

Step 9.2.2

Multiply $- 2$ by $1$ .

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

Step 9.2.3

Subtract $2$ from $1$ .

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

Step 9.3

Divide $- 2$ by $- 1$ .

$2$