Calculus Examples

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

Step 1

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

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

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.

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

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 $0$ .

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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Step 2.1.2.3.1

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

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

Step 2.1.2.3.2

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

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

Step 2.1.2.4

Simplify the answer.

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Step 2.1.2.4.1

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

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

Step 2.1.2.4.2

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

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

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

Step 2.3.3

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

$\frac{1}{2} lim_{x \to 0} \frac{2 x + \frac{d}{d x} [x]}{\frac{d}{d x} [x]}$

Step 2.3.4

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

$\frac{1}{2} lim_{x \to 0} \frac{2 x + 1}{\frac{d}{d x} [x]}$

Step 2.3.5

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

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

Step 2.4

Divide $2 x + 1$ by $1$ .

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

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 $0$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

Multiply $2$ by $0$ .

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

Step 5.2

Add $0$ and $1$ .

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

Step 5.3

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

$\frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$