Calculus Examples

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

Step 1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} - x + \sin (x)}{x}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 2.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{1}{2} \cdot \frac{lim_{x \to 0} x^{2} - x + \sin (x)}{lim_{x \to 0} x}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

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

$\frac{1}{2} \cdot \frac{lim_{x \to 0} x^{2} - lim_{x \to 0} x + lim_{x \to 0} \sin (x)}{lim_{x \to 0} x}$

Step 2.1.2.2

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

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} - lim_{x \to 0} x + lim_{x \to 0} \sin (x)}{lim_{x \to 0} x}$

Step 2.1.2.3

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

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} - lim_{x \to 0} x + \sin (lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 2.1.2.4

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

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Step 2.1.2.4.1

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

$\frac{1}{2} \cdot \frac{0^{2} - lim_{x \to 0} x + \sin (lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 2.1.2.4.2

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

$\frac{1}{2} \cdot \frac{0^{2} + 0 + \sin (lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 2.1.2.4.3

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

$\frac{1}{2} \cdot \frac{0^{2} + 0 + \sin (0)}{lim_{x \to 0} x}$

Step 2.1.2.5

Simplify the answer.

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Step 2.1.2.5.1

Simplify each term.

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Step 2.1.2.5.1.1

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

$\frac{1}{2} \cdot \frac{0 + 0 + \sin (0)}{lim_{x \to 0} x}$

Step 2.1.2.5.1.2

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

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

Step 2.1.2.5.2

Add $0$ and $0$ .

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

Step 2.1.2.5.3

Add $0$ and $0$ .

$\frac{1}{2} \cdot \frac{0}{lim_{x \to 0} x}$

Step 2.1.3

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

$\frac{1}{2} \cdot \frac{0}{0}$

Step 2.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

Step 2.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$\frac{1}{2} lim_{x \to 0} \frac{\frac{d}{d x} [x^{2} - x + \sin (x)]}{\frac{d}{d x} [x]}$

Step 2.3.2

By the Sum Rule, the derivative of $x^{2} - x + \sin (x)$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [\sin (x)]$ .

$\frac{1}{2} lim_{x \to 0} \frac{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 2.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 x + \frac{d}{d x} [- x] + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 2.3.4

Evaluate $\frac{d}{d x} [- x]$ .

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Step 2.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 x - \frac{d}{d x} [x] + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 2.3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 x - 1 \cdot 1 + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 2.3.4.3

Multiply $- 1$ by $1$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 x - 1 + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 2.3.5

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 x - 1 + \cos (x)}{\frac{d}{d x} [x]}$

Step 2.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

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

Step 2.4

Divide $2 x - 1 + \cos (x)$ by $1$ .

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

Step 3

Evaluate the limit.

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Step 3.1

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

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

Step 3.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 3.3

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

$\frac{1}{2} (2 lim_{x \to 0} x - 1 \cdot 1 + lim_{x \to 0} \cos (x))$

Step 3.4

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

$\frac{1}{2} (2 lim_{x \to 0} x - 1 \cdot 1 + \cos (lim_{x \to 0} x))$

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 $x$ .

$\frac{1}{2} (2 \cdot 0 - 1 \cdot 1 + \cos (lim_{x \to 0} x))$

Step 4.2

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

$\frac{1}{2} (2 \cdot 0 - 1 \cdot 1 + \cos (0))$

Step 5

Simplify the answer.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $2$ by $0$ .

$\frac{1}{2} (0 - 1 \cdot 1 + \cos (0))$

Step 5.1.2

Multiply $- 1$ by $1$ .

$\frac{1}{2} (0 - 1 + \cos (0))$

Step 5.1.3

The exact value of $\cos (0)$ is $1$ .

$\frac{1}{2} (0 - 1 + 1)$

Step 5.2

Subtract $1$ from $0$ .

$\frac{1}{2} (- 1 + 1)$

Step 5.3

Add $- 1$ and $1$ .

$\frac{1}{2} \cdot 0$

Step 5.4

Multiply $\frac{1}{2}$ by $0$ .

$0$