Calculus Examples

$\frac{lim_{x \to 1} x^{3} - x^{2} + x - 1}{x^{4} - x^{3} - 2 x + 2}$

Step 1

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

$\frac{lim_{x \to 1} x^{3} - lim_{x \to 1} x^{2} + lim_{x \to 1} x - lim_{x \to 1} 1}{x^{4} - x^{3} - 2 x + 2}$

Step 2

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 1} x)}^{3} - lim_{x \to 1} x^{2} + lim_{x \to 1} x - lim_{x \to 1} 1}{x^{4} - x^{3} - 2 x + 2}$

Step 3

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

$\frac{{(lim_{x \to 1} x)}^{3} - {(lim_{x \to 1} x)}^{2} + lim_{x \to 1} x - lim_{x \to 1} 1}{x^{4} - x^{3} - 2 x + 2}$

Step 4

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

$\frac{{(lim_{x \to 1} x)}^{3} - {(lim_{x \to 1} x)}^{2} + lim_{x \to 1} x - 1 \cdot 1}{x^{4} - x^{3} - 2 x + 2}$

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 $x$ .

$\frac{1^{3} - {(lim_{x \to 1} x)}^{2} + lim_{x \to 1} x - 1 \cdot 1}{x^{4} - x^{3} - 2 x + 2}$

Step 5.2

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

$\frac{1^{3} - 1^{2} + lim_{x \to 1} x - 1 \cdot 1}{x^{4} - x^{3} - 2 x + 2}$

Step 5.3

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

$\frac{1^{3} - 1^{2} + 1 - 1 \cdot 1}{x^{4} - x^{3} - 2 x + 2}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

One to any power is one.

$\frac{1 - 1^{2} + 1 - 1 \cdot 1}{x^{4} - x^{3} - 2 x + 2}$

Step 6.1.2

One to any power is one.

$\frac{1 - 1 \cdot 1 + 1 - 1 \cdot 1}{x^{4} - x^{3} - 2 x + 2}$

Step 6.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - 1 + 1 - 1 \cdot 1}{x^{4} - x^{3} - 2 x + 2}$

Step 6.1.4

Multiply $- 1$ by $1$ .

$\frac{1 - 1 + 1 - 1}{x^{4} - x^{3} - 2 x + 2}$

Step 6.1.5

Subtract $1$ from $1$ .

$\frac{0 + 1 - 1}{x^{4} - x^{3} - 2 x + 2}$

Step 6.1.6

Add $0$ and $1$ .

$\frac{1 - 1}{x^{4} - x^{3} - 2 x + 2}$

Step 6.1.7

Subtract $1$ from $1$ .

$\frac{0}{x^{4} - x^{3} - 2 x + 2}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Factor out the greatest common factor from each group.

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Step 6.2.1.1

Group the first two terms and the last two terms.

$\frac{0}{(x^{4} - x^{3}) - 2 x + 2}$

Step 6.2.1.2

Factor out the greatest common factor (GCF) from each group.

$\frac{0}{x^{3} (x - 1) - 2 (x - 1)}$

Step 6.2.2

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$\frac{0}{(x - 1) (x^{3} - 2)}$

Step 6.3

Divide $0$ by $(x - 1) (x^{3} - 2)$ .

$0$