Calculus Examples

$\frac{lim_{x \to 0} x^{5} + 3 x}{x}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} x^{5} + lim_{x \to 0} 3 x}{x}$

Step 2

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 0} x)}^{5} + lim_{x \to 0} 3 x}{x}$

Step 3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 0} x)}^{5} + 3 lim_{x \to 0} x}{x}$

Step 4

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 4.1

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

$\frac{0^{5} + 3 lim_{x \to 0} x}{x}$

Step 4.2

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

$\frac{0^{5} + 3 \cdot 0}{x}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$\frac{0 + 3 \cdot 0}{x}$

Step 5.1.2

Multiply $3$ by $0$ .

$\frac{0 + 0}{x}$

Step 5.1.3

Add $0$ and $0$ .

$\frac{0}{x}$

Step 5.2

Divide $0$ by $x$ .

$0$