Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

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 $x$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

Simplify the answer.

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Step 6.1

Multiply $1^{3} - 1^{2}$ by $1$ .

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

Step 6.2

Simplify the numerator.

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Step 6.2.1

One to any power is one.

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

Step 6.2.2

One to any power is one.

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

Step 6.2.3

Multiply $- 1$ by $1$ .

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

Step 6.2.4

Subtract $1$ from $1$ .

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

Step 6.3

Divide $0$ by $x - 1$ .

$0$