Calculus Examples

$lim_{x \to 0} \frac{\frac{3}{x + 3} - \frac{3}{3}}{3 x}$

Step 1

Reduce the expression by cancelling the common factors.

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Step 1.1

Cancel the common factor of $3$ .

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Step 1.1.1

Cancel the common factor.

$lim_{x \to 0} \frac{\frac{3}{x + 3} - \frac{3}{3}}{3 x}$

Step 1.1.2

Rewrite the expression.

$lim_{x \to 0} \frac{\frac{3}{x + 3} - 1 \cdot 1}{3 x}$

Step 1.2

Multiply $- 1$ by $1$ .

$lim_{x \to 0} \frac{\frac{3}{x + 3} - 1}{3 x}$

Step 2

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} lim_{x \to 0} \frac{\frac{3}{x + 3} - 1}{x}$

Step 3

Combine terms.

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Step 3.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{1}{3} lim_{x \to 0} \frac{\frac{3}{x + 3} - 1 \cdot \frac{x + 3}{x + 3}}{x}$

Step 3.2

Combine $- 1$ and $\frac{x + 3}{x + 3}$ .

$\frac{1}{3} lim_{x \to 0} \frac{\frac{3}{x + 3} + \frac{- (x + 3)}{x + 3}}{x}$

Step 3.3

Combine the numerators over the common denominator.

$\frac{1}{3} lim_{x \to 0} \frac{\frac{3 - (x + 3)}{x + 3}}{x}$

Step 4

Simplify the limit argument.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{3} lim_{x \to 0} \frac{3 - (x + 3)}{x + 3} \cdot \frac{1}{x}$

Step 4.2

Multiply $\frac{3 - (x + 3)}{x + 3}$ by $\frac{1}{x}$ .

$\frac{1}{3} lim_{x \to 0} \frac{3 - (x + 3)}{(x + 3) x}$

Step 5

Apply L'Hospital's rule.

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Step 5.1

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

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Step 5.1.1

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

$\frac{1}{3} \cdot \frac{lim_{x \to 0} 3 - (x + 3)}{lim_{x \to 0} (x + 3) x}$

Step 5.1.2

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

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Step 5.1.2.1

Evaluate the limit of $3 - (x + 3)$ by plugging in $0$ for $x$ .

$\frac{1}{3} \cdot \frac{3 - (0 + 3)}{lim_{x \to 0} (x + 3) x}$

Step 5.1.2.2

Simplify each term.

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Step 5.1.2.2.1

Add $0$ and $3$ .

$\frac{1}{3} \cdot \frac{3 - 1 \cdot 3}{lim_{x \to 0} (x + 3) x}$

Step 5.1.2.2.2

Multiply $- 1$ by $3$ .

$\frac{1}{3} \cdot \frac{3 - 3}{lim_{x \to 0} (x + 3) x}$

Step 5.1.2.3

Subtract $3$ from $3$ .

$\frac{1}{3} \cdot \frac{0}{lim_{x \to 0} (x + 3) x}$

Step 5.1.3

Evaluate the limit of the denominator.

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Step 5.1.3.1

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

$\frac{1}{3} \cdot \frac{0}{lim_{x \to 0} x + 3 \cdot lim_{x \to 0} x}$

Step 5.1.3.2

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

$\frac{1}{3} \cdot \frac{0}{(lim_{x \to 0} x + lim_{x \to 0} 3) \cdot lim_{x \to 0} x}$

Step 5.1.3.3

Evaluate the limit of $3$ which is constant as $x$ approaches $0$ .

$\frac{1}{3} \cdot \frac{0}{(lim_{x \to 0} x + 3) \cdot lim_{x \to 0} x}$

Step 5.1.3.4

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

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Step 5.1.3.4.1

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

$\frac{1}{3} \cdot \frac{0}{(0 + 3) \cdot lim_{x \to 0} x}$

Step 5.1.3.4.2

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

$\frac{1}{3} \cdot \frac{0}{(0 + 3) \cdot 0}$

Step 5.1.3.5

Simplify the answer.

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Step 5.1.3.5.1

Add $0$ and $3$ .

$\frac{1}{3} \cdot \frac{0}{3 \cdot 0}$

Step 5.1.3.5.2

Multiply $3$ by $0$ .

$\frac{1}{3} \cdot \frac{0}{0}$

Step 5.1.3.5.3

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

Undefined

$\frac{1}{3} \cdot \frac{0}{0}$

Step 5.1.3.6

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

Undefined

$\frac{1}{3} \cdot \frac{0}{0}$

Step 5.1.4

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

Undefined

$\frac{1}{3} \cdot \frac{0}{0}$

Step 5.2

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.

$lim_{x \to 0} \frac{3 - (x + 3)}{(x + 3) x} = lim_{x \to 0} \frac{\frac{d}{d x} [3 - (x + 3)]}{\frac{d}{d x} [(x + 3) x]}$

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

$\frac{1}{3} lim_{x \to 0} \frac{\frac{d}{d x} [3 - (x + 3)]}{\frac{d}{d x} [(x + 3) x]}$

Step 5.3.2

By the Sum Rule, the derivative of $3 - (x + 3)$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- (x + 3)]$ .

$\frac{1}{3} lim_{x \to 0} \frac{\frac{d}{d x} [3] + \frac{d}{d x} [- (x + 3)]}{\frac{d}{d x} [(x + 3) x]}$

Step 5.3.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{1}{3} lim_{x \to 0} \frac{0 + \frac{d}{d x} [- (x + 3)]}{\frac{d}{d x} [(x + 3) x]}$

Step 5.3.4

Evaluate $\frac{d}{d x} [- (x + 3)]$ .

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Step 5.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (x + 3)$ with respect to $x$ is $- \frac{d}{d x} [x + 3]$ .

$\frac{1}{3} lim_{x \to 0} \frac{0 - \frac{d}{d x} [x + 3]}{\frac{d}{d x} [(x + 3) x]}$

Step 5.3.4.2

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$\frac{1}{3} lim_{x \to 0} \frac{0 - (\frac{d}{d x} [x] + \frac{d}{d x} [3])}{\frac{d}{d x} [(x + 3) x]}$

Step 5.3.4.3

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

$\frac{1}{3} lim_{x \to 0} \frac{0 - (1 + \frac{d}{d x} [3])}{\frac{d}{d x} [(x + 3) x]}$

Step 5.3.4.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{1}{3} lim_{x \to 0} \frac{0 - (1 + 0)}{\frac{d}{d x} [(x + 3) x]}$

Step 5.3.4.5

Add $1$ and $0$ .

$\frac{1}{3} lim_{x \to 0} \frac{0 - 1 \cdot 1}{\frac{d}{d x} [(x + 3) x]}$

Step 5.3.4.6

Multiply $- 1$ by $1$ .

$\frac{1}{3} lim_{x \to 0} \frac{0 - 1}{\frac{d}{d x} [(x + 3) x]}$

Step 5.3.5

Subtract $1$ from $0$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 1}{\frac{d}{d x} [(x + 3) x]}$

Step 5.3.6

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x + 3$ and $g (x) = x$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 1}{(x + 3) \frac{d}{d x} [x] + x \frac{d}{d x} [x + 3]}$

Step 5.3.7

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

$\frac{1}{3} lim_{x \to 0} \frac{- 1}{(x + 3) \cdot 1 + x \frac{d}{d x} [x + 3]}$

Step 5.3.8

Multiply $x + 3$ by $1$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 1}{x + 3 + x \frac{d}{d x} [x + 3]}$

Step 5.3.9

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 1}{x + 3 + x (\frac{d}{d x} [x] + \frac{d}{d x} [3])}$

Step 5.3.10

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

$\frac{1}{3} lim_{x \to 0} \frac{- 1}{x + 3 + x (1 + \frac{d}{d x} [3])}$

Step 5.3.11

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 1}{x + 3 + x (1 + 0)}$

Step 5.3.12

Add $1$ and $0$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 1}{x + 3 + x \cdot 1}$

Step 5.3.13

Multiply $x$ by $1$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 1}{x + 3 + x}$

Step 5.3.14

Add $x$ and $x$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 1}{2 x + 3}$

Step 6

Evaluate the limit.

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Step 6.1

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

$\frac{1}{3} \cdot \frac{lim_{x \to 0} - 1}{lim_{x \to 0} 2 x + 3}$

Step 6.2

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

$\frac{1}{3} \cdot \frac{- 1}{lim_{x \to 0} 2 x + 3}$

Step 6.3

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

$\frac{1}{3} \cdot \frac{- 1}{lim_{x \to 0} 2 x + lim_{x \to 0} 3}$

Step 6.4

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

$\frac{1}{3} \cdot \frac{- 1}{2 lim_{x \to 0} x + lim_{x \to 0} 3}$

Step 6.5

Evaluate the limit of $3$ which is constant as $x$ approaches $0$ .

$\frac{1}{3} \cdot \frac{- 1}{2 lim_{x \to 0} x + 3}$

Step 7

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

$\frac{1}{3} \cdot \frac{- 1}{2 \cdot 0 + 3}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the denominator.

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Step 8.1.1

Multiply $2$ by $0$ .

$\frac{1}{3} \cdot \frac{- 1}{0 + 3}$

Step 8.1.2

Add $0$ and $3$ .

$\frac{1}{3} \cdot \frac{- 1}{3}$

Step 8.2

Move the negative in front of the fraction.

$\frac{1}{3} (- \frac{1}{3})$

Step 8.3

Multiply $\frac{1}{3} (- \frac{1}{3})$ .

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Step 8.3.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$- \frac{1}{3 \cdot 3}$

Step 8.3.2

Multiply $3$ by $3$ .

$- \frac{1}{9}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{9}$

Decimal Form:

$- 0.\overline{1}$