Calculus Examples

$lim_{x \to 0} \frac{3 x + 2 x^{2}}{5 x + 6 x^{3}}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{x \to 0} 3 x + 2 x^{2}}{lim_{x \to 0} 5 x + 6 x^{3}}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

$\frac{lim_{x \to 0} 3 x + lim_{x \to 0} 2 x^{2}}{lim_{x \to 0} 5 x + 6 x^{3}}$

Step 1.1.2.2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 lim_{x \to 0} x + lim_{x \to 0} 2 x^{2}}{lim_{x \to 0} 5 x + 6 x^{3}}$

Step 1.1.2.3

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

$\frac{3 lim_{x \to 0} x + 2 lim_{x \to 0} x^{2}}{lim_{x \to 0} 5 x + 6 x^{3}}$

Step 1.1.2.4

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

$\frac{3 lim_{x \to 0} x + 2 {(lim_{x \to 0} x)}^{2}}{lim_{x \to 0} 5 x + 6 x^{3}}$

Step 1.1.2.5

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

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Step 1.1.2.5.1

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

$\frac{3 \cdot 0 + 2 {(lim_{x \to 0} x)}^{2}}{lim_{x \to 0} 5 x + 6 x^{3}}$

Step 1.1.2.5.2

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

$\frac{3 \cdot 0 + 2 \cdot 0^{2}}{lim_{x \to 0} 5 x + 6 x^{3}}$

Step 1.1.2.6

Simplify the answer.

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Step 1.1.2.6.1

Simplify each term.

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Step 1.1.2.6.1.1

Multiply $3$ by $0$ .

$\frac{0 + 2 \cdot 0^{2}}{lim_{x \to 0} 5 x + 6 x^{3}}$

Step 1.1.2.6.1.2

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

$\frac{0 + 2 \cdot 0}{lim_{x \to 0} 5 x + 6 x^{3}}$

Step 1.1.2.6.1.3

Multiply $2$ by $0$ .

$\frac{0 + 0}{lim_{x \to 0} 5 x + 6 x^{3}}$

Step 1.1.2.6.2

Add $0$ and $0$ .

$\frac{0}{lim_{x \to 0} 5 x + 6 x^{3}}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

$\frac{0}{lim_{x \to 0} 5 x + lim_{x \to 0} 6 x^{3}}$

Step 1.1.3.2

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{5 lim_{x \to 0} x + lim_{x \to 0} 6 x^{3}}$

Step 1.1.3.3

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{5 lim_{x \to 0} x + 6 lim_{x \to 0} x^{3}}$

Step 1.1.3.4

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{0}{5 lim_{x \to 0} x + 6 {(lim_{x \to 0} x)}^{3}}$

Step 1.1.3.5

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

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Step 1.1.3.5.1

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

$\frac{0}{5 \cdot 0 + 6 {(lim_{x \to 0} x)}^{3}}$

Step 1.1.3.5.2

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

$\frac{0}{5 \cdot 0 + 6 \cdot 0^{3}}$

Step 1.1.3.6

Simplify the answer.

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Step 1.1.3.6.1

Simplify each term.

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Step 1.1.3.6.1.1

Multiply $5$ by $0$ .

$\frac{0}{0 + 6 \cdot 0^{3}}$

Step 1.1.3.6.1.2

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

$\frac{0}{0 + 6 \cdot 0}$

Step 1.1.3.6.1.3

Multiply $6$ by $0$ .

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

Step 1.1.3.6.2

Add $0$ and $0$ .

$\frac{0}{0}$

Step 1.1.3.6.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.7

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

$lim_{x \to 0} \frac{3 x + 2 x^{2}}{5 x + 6 x^{3}} = lim_{x \to 0} \frac{\frac{d}{d x} [3 x + 2 x^{2}]}{\frac{d}{d x} [5 x + 6 x^{3}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [3 x + 2 x^{2}]}{\frac{d}{d x} [5 x + 6 x^{3}]}$

Step 1.3.2

By the Sum Rule, the derivative of $3 x + 2 x^{2}$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [2 x^{2}]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [3 x] + \frac{d}{d x} [2 x^{2}]}{\frac{d}{d x} [5 x + 6 x^{3}]}$

Step 1.3.3

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 1.3.3.1

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

$lim_{x \to 0} \frac{3 \frac{d}{d x} [x] + \frac{d}{d x} [2 x^{2}]}{\frac{d}{d x} [5 x + 6 x^{3}]}$

Step 1.3.3.2

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

$lim_{x \to 0} \frac{3 \cdot 1 + \frac{d}{d x} [2 x^{2}]}{\frac{d}{d x} [5 x + 6 x^{3}]}$

Step 1.3.3.3

Multiply $3$ by $1$ .

$lim_{x \to 0} \frac{3 + \frac{d}{d x} [2 x^{2}]}{\frac{d}{d x} [5 x + 6 x^{3}]}$

Step 1.3.4

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

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Step 1.3.4.1

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

$lim_{x \to 0} \frac{3 + 2 \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [5 x + 6 x^{3}]}$

Step 1.3.4.2

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

$lim_{x \to 0} \frac{3 + 2 (2 x)}{\frac{d}{d x} [5 x + 6 x^{3}]}$

Step 1.3.4.3

Multiply $2$ by $2$ .

$lim_{x \to 0} \frac{3 + 4 x}{\frac{d}{d x} [5 x + 6 x^{3}]}$

Step 1.3.5

Reorder terms.

$lim_{x \to 0} \frac{4 x + 3}{\frac{d}{d x} [5 x + 6 x^{3}]}$

Step 1.3.6

By the Sum Rule, the derivative of $5 x + 6 x^{3}$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [6 x^{3}]$ .

$lim_{x \to 0} \frac{4 x + 3}{\frac{d}{d x} [5 x] + \frac{d}{d x} [6 x^{3}]}$

Step 1.3.7

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 1.3.7.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{4 x + 3}{5 \frac{d}{d x} [x] + \frac{d}{d x} [6 x^{3}]}$

Step 1.3.7.2

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

$lim_{x \to 0} \frac{4 x + 3}{5 \cdot 1 + \frac{d}{d x} [6 x^{3}]}$

Step 1.3.7.3

Multiply $5$ by $1$ .

$lim_{x \to 0} \frac{4 x + 3}{5 + \frac{d}{d x} [6 x^{3}]}$

Step 1.3.8

Evaluate $\frac{d}{d x} [6 x^{3}]$ .

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Step 1.3.8.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{3}$ with respect to $x$ is $6 \frac{d}{d x} [x^{3}]$ .

$lim_{x \to 0} \frac{4 x + 3}{5 + 6 \frac{d}{d x} [x^{3}]}$

Step 1.3.8.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$lim_{x \to 0} \frac{4 x + 3}{5 + 6 (3 x^{2})}$

Step 1.3.8.3

Multiply $3$ by $6$ .

$lim_{x \to 0} \frac{4 x + 3}{5 + 18 x^{2}}$

Step 1.3.9

Reorder terms.

$lim_{x \to 0} \frac{4 x + 3}{18 x^{2} + 5}$

Step 2

Evaluate the limit.

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Step 2.1

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

$\frac{lim_{x \to 0} 4 x + 3}{lim_{x \to 0} 18 x^{2} + 5}$

Step 2.2

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

$\frac{lim_{x \to 0} 4 x + lim_{x \to 0} 3}{lim_{x \to 0} 18 x^{2} + 5}$

Step 2.3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 lim_{x \to 0} x + lim_{x \to 0} 3}{lim_{x \to 0} 18 x^{2} + 5}$

Step 2.4

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

$\frac{4 lim_{x \to 0} x + 3}{lim_{x \to 0} 18 x^{2} + 5}$

Step 2.5

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

$\frac{4 lim_{x \to 0} x + 3}{lim_{x \to 0} 18 x^{2} + lim_{x \to 0} 5}$

Step 2.6

Move the term $18$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 lim_{x \to 0} x + 3}{18 lim_{x \to 0} x^{2} + lim_{x \to 0} 5}$

Step 2.7

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

$\frac{4 lim_{x \to 0} x + 3}{18 {(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 5}$

Step 2.8

Evaluate the limit of $5$ which is constant as $x$ approaches $0$ .

$\frac{4 lim_{x \to 0} x + 3}{18 {(lim_{x \to 0} x)}^{2} + 5}$

Step 3

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

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Step 3.1

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

$\frac{4 \cdot 0 + 3}{18 {(lim_{x \to 0} x)}^{2} + 5}$

Step 3.2

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

$\frac{4 \cdot 0 + 3}{18 \cdot 0^{2} + 5}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $4$ by $0$ .

$\frac{0 + 3}{18 \cdot 0^{2} + 5}$

Step 4.1.2

Add $0$ and $3$ .

$\frac{3}{18 \cdot 0^{2} + 5}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

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

$\frac{3}{18 \cdot 0 + 5}$

Step 4.2.2

Multiply $18$ by $0$ .

$\frac{3}{0 + 5}$

Step 4.2.3

Add $0$ and $5$ .

$\frac{3}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{5}$

Decimal Form:

$0.6$