Calculus Examples

$lim_{x \to 0} x (13 + \frac{3}{x})$

Step 1

Combine terms.

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Step 1.1

To write $13$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$lim_{x \to 0} x (\frac{13 x}{x} + \frac{3}{x})$

Step 1.2

Combine the numerators over the common denominator.

$lim_{x \to 0} x \frac{13 x + 3}{x}$

Step 2

Evaluate the limit.

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Step 2.1

Simplify the limit argument.

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Step 2.1.1

Combine $x$ and $\frac{13 x + 3}{x}$ .

$lim_{x \to 0} \frac{x (13 x + 3)}{x}$

Step 2.1.2

Cancel the common factor of $x$ .

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Step 2.1.2.1

Cancel the common factor.

$lim_{x \to 0} \frac{x (13 x + 3)}{x}$

Step 2.1.2.2

Divide $13 x + 3$ by $1$ .

$lim_{x \to 0} 13 x + 3$

Step 2.2

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

$lim_{x \to 0} 13 x + lim_{x \to 0} 3$

Step 2.3

Move the term $13$ outside of the limit because it is constant with respect to $x$ .

$13 lim_{x \to 0} x + lim_{x \to 0} 3$

Step 2.4

Evaluate the limit of $3$ which is constant as $x$ approaches $0$ .

$13 lim_{x \to 0} x + 3$

Step 3

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

$13 \cdot 0 + 3$

Step 4

Simplify the answer.

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Step 4.1

Multiply $13$ by $0$ .

$0 + 3$

Step 4.2

Add $0$ and $3$ .

$3$