Calculus Examples

$\frac{lim_{x \to 0} \frac{1}{2 + x}}{x} - \frac{\frac{1}{2}}{x}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{\frac{lim_{x \to 0} 1}{lim_{x \to 0} 2 + x}}{x} - \frac{\frac{1}{2}}{x}$

Step 1.2

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

$\frac{\frac{1}{lim_{x \to 0} 2 + x}}{x} - \frac{\frac{1}{2}}{x}$

Step 1.3

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

$\frac{\frac{1}{lim_{x \to 0} 2 + lim_{x \to 0} x}}{x} - \frac{\frac{1}{2}}{x}$

Step 1.4

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$\frac{\frac{1}{2 + lim_{x \to 0} x}}{x} - \frac{\frac{1}{2}}{x}$

Step 2

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

$\frac{\frac{1}{2 + 0}}{x} - \frac{\frac{1}{2}}{x}$

Step 3

Combine the opposite terms in $\frac{\frac{1}{2 + 0}}{x} - \frac{\frac{1}{2}}{x}$ .

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Step 3.1

Add $2$ and $0$ .

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

Step 3.2

Subtract $\frac{\frac{1}{2}}{x}$ from $\frac{\frac{1}{2}}{x}$ .

$0$