Calculus Examples

$\frac{lim_{x \to 0} x^{2} - x + \sin (x)}{2 x}$

Step 1

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

$\frac{lim_{x \to 0} x^{2} - lim_{x \to 0} x + lim_{x \to 0} \sin (x)}{2 x}$

Step 2

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

$\frac{{(lim_{x \to 0} x)}^{2} - lim_{x \to 0} x + lim_{x \to 0} \sin (x)}{2 x}$

Step 3

Move the limit inside the trig function because sine is continuous.

$\frac{{(lim_{x \to 0} x)}^{2} - lim_{x \to 0} x + \sin (lim_{x \to 0} x)}{2 x}$

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 $x$ .

$\frac{0^{2} - lim_{x \to 0} x + \sin (lim_{x \to 0} x)}{2 x}$

Step 4.2

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

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

Step 4.3

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

$\frac{0^{2} + 0 + \sin (0)}{2 x}$

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 $0$ .

$\frac{0 + 0 + \sin (0)}{2 x}$

Step 5.1.2

The exact value of $\sin (0)$ is $0$ .

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

Step 5.1.3

Add $0 + 0$ and $0$ .

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

Step 5.1.4

Add $0$ and $0$ .

$\frac{0}{2 x}$

Step 5.2

Cancel the common factor of $0$ and $2$ .

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Step 5.2.1

Factor $2$ out of $0$ .

$\frac{2 (0)}{2 x}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $2 x$ .

$\frac{2 (0)}{2 (x)}$

Step 5.2.2.2

Cancel the common factor.

$\frac{2 \cdot 0}{2 x}$

Step 5.2.2.3

Rewrite the expression.

$\frac{0}{x}$

Step 5.3

Divide $0$ by $x$ .

$0$