Calculus Examples

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

Step 1

Simplify the limit argument.

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Step 1.1

Combine terms.

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Step 1.1.1

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + 2 x}{x^{2} + 2 x}$ .

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

Step 1.1.2

To write $- \frac{2}{x^{2} + 2 x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 1.1.3

Write each expression with a common denominator of $x (x^{2} + 2 x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.3.1

Multiply $\frac{1}{x}$ by $\frac{x^{2} + 2 x}{x^{2} + 2 x}$ .

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

Step 1.1.3.2

Multiply $\frac{2}{x^{2} + 2 x}$ by $\frac{x}{x}$ .

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

Step 1.1.3.3

Reorder the factors of $(x^{2} + 2 x) x$ .

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

Step 1.1.4

Combine the numerators over the common denominator.

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

Step 1.1.5

Subtract $2 x$ from $2 x$ .

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

Step 1.1.6

Add $x^{2}$ and $0$ .

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

Step 1.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 1.2.1

Factor $x$ out of $x^{2}$ .

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

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Cancel the common factor.

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

Step 1.2.2.2

Rewrite the expression.

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

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.

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

Step 2.1.2

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

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

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

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

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Step 2.1.3.4.1

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

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

Step 2.1.3.4.2

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

$\frac{0}{0^{2} + 2 \cdot 0}$

Step 2.1.3.5

Simplify the answer.

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Step 2.1.3.5.1

Simplify each term.

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Step 2.1.3.5.1.1

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

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

Step 2.1.3.5.1.2

Multiply $2$ by $0$ .

$\frac{0}{0 + 0}$

Step 2.1.3.5.2

Add $0$ and $0$ .

$\frac{0}{0}$

Step 2.1.3.5.3

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

Undefined

$\frac{0}{0}$

Step 2.1.3.6

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

Undefined

$\frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{0}{0}$

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 2.3.5.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 2.3.5.2

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

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

Step 2.3.5.3

Multiply $2$ by $1$ .

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

Step 3

Evaluate the limit.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 4

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

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

Step 5

Simplify the denominator.

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Step 5.1

Multiply $2$ by $0$ .

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

Step 5.2

Add $0$ and $2$ .

$\frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$