Calculus Examples

$lim_{x \to 0} x (1 + \frac{1}{x})$

Step 1

Combine terms.

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Step 1.1

Write $1$ as a fraction with a common denominator.

$lim_{x \to 0} x (\frac{x}{x} + \frac{1}{x})$

Step 1.2

Combine the numerators over the common denominator.

$lim_{x \to 0} x \frac{x + 1}{x}$

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 $\frac{x + 1}{x}$ .

$lim_{x \to 0} \frac{x (x + 1)}{x}$

Step 2.1.2

Cancel the common factor of $x$ .

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Step 2.1.2.1

Cancel the common factor.

$lim_{x \to 0} \frac{x (x + 1)}{x}$

Step 2.1.2.2

Divide $x + 1$ by $1$ .

$lim_{x \to 0} x + 1$

Step 2.2

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

$lim_{x \to 0} x + lim_{x \to 0} 1$

Step 2.3

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$lim_{x \to 0} x + 1$

Step 3

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

$0 + 1$

Step 4

Add $0$ and $1$ .

$1$