Calculus Examples

$lim_{x \to 0} \frac{x^{3}}{x}$

Step 1

Evaluate the limit.

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Step 1.1

Cancel the common factor of $x^{3}$ and $x$ .

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Step 1.1.1

Factor $x$ out of $x^{3}$ .

$lim_{x \to 0} \frac{x \cdot x^{2}}{x}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Raise $x$ to the power of $1$ .

$lim_{x \to 0} \frac{x \cdot x^{2}}{x^{1}}$

Step 1.1.2.2

Factor $x$ out of $x^{1}$ .

$lim_{x \to 0} \frac{x \cdot x^{2}}{x \cdot 1}$

Step 1.1.2.3

Cancel the common factor.

$lim_{x \to 0} \frac{x \cdot x^{2}}{x \cdot 1}$

Step 1.1.2.4

Rewrite the expression.

$lim_{x \to 0} \frac{x^{2}}{1}$

Step 1.1.2.5

Divide $x^{2}$ by $1$ .

$lim_{x \to 0} x^{2}$

Step 1.2

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

${(lim_{x \to 0} x)}^{2}$

Step 2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$0^{2}$

Step 3

Raising $0$ to any positive power yields $0$ .

$0$