Calculus Examples

$lim_{x \to 0} \frac{4 \ln (2 x + 1) + x^{3}}{5 x^{2} + 3 x}$

Step 1

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

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Step 1.1

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

$\frac{lim_{x \to 0} 4 \ln (2 x + 1) + x^{3}}{lim_{x \to 0} 5 x^{2} + 3 x}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

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

$\frac{lim_{x \to 0} 4 \ln (2 x + 1) + lim_{x \to 0} x^{3}}{lim_{x \to 0} 5 x^{2} + 3 x}$

Step 1.2.2

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

$\frac{4 lim_{x \to 0} \ln (2 x + 1) + lim_{x \to 0} x^{3}}{lim_{x \to 0} 5 x^{2} + 3 x}$

Step 1.2.3

Move the limit inside the logarithm.

$\frac{4 \ln (lim_{x \to 0} 2 x + 1) + lim_{x \to 0} x^{3}}{lim_{x \to 0} 5 x^{2} + 3 x}$

Step 1.2.4

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

$\frac{4 \ln (lim_{x \to 0} 2 x + lim_{x \to 0} 1) + lim_{x \to 0} x^{3}}{lim_{x \to 0} 5 x^{2} + 3 x}$

Step 1.2.5

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

$\frac{4 \ln (2 lim_{x \to 0} x + lim_{x \to 0} 1) + lim_{x \to 0} x^{3}}{lim_{x \to 0} 5 x^{2} + 3 x}$

Step 1.2.6

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

$\frac{4 \ln (2 lim_{x \to 0} x + 1) + lim_{x \to 0} x^{3}}{lim_{x \to 0} 5 x^{2} + 3 x}$

Step 1.2.7

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

$\frac{4 \ln (2 lim_{x \to 0} x + 1) + {(lim_{x \to 0} x)}^{3}}{lim_{x \to 0} 5 x^{2} + 3 x}$

Step 1.2.8

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

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Step 1.2.8.1

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

$\frac{4 \ln (2 \cdot 0 + 1) + {(lim_{x \to 0} x)}^{3}}{lim_{x \to 0} 5 x^{2} + 3 x}$

Step 1.2.8.2

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

$\frac{4 \ln (2 \cdot 0 + 1) + 0^{3}}{lim_{x \to 0} 5 x^{2} + 3 x}$

Step 1.2.9

Simplify the answer.

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Step 1.2.9.1

Simplify each term.

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Step 1.2.9.1.1

Multiply $2$ by $0$ .

$\frac{4 \ln (0 + 1) + 0^{3}}{lim_{x \to 0} 5 x^{2} + 3 x}$

Step 1.2.9.1.2

Add $0$ and $1$ .

$\frac{4 \ln (1) + 0^{3}}{lim_{x \to 0} 5 x^{2} + 3 x}$

Step 1.2.9.1.3

The natural logarithm of $1$ is $0$ .

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

Step 1.2.9.1.4

Multiply $4$ by $0$ .

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

Step 1.2.9.1.5

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

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

Step 1.2.9.2

Add $0$ and $0$ .

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

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

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Step 1.3.5.1

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

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

Step 1.3.5.2

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

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

Step 1.3.6

Simplify the answer.

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Step 1.3.6.1

Simplify each term.

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Step 1.3.6.1.1

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

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

Step 1.3.6.1.2

Multiply $5$ by $0$ .

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

Step 1.3.6.1.3

Multiply $3$ by $0$ .

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

Step 1.3.6.2

Add $0$ and $0$ .

$\frac{0}{0}$

Step 1.3.6.3

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

Undefined

$\frac{0}{0}$

Step 1.3.7

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

$lim_{x \to 0} \frac{4 \ln (2 x + 1) + x^{3}}{5 x^{2} + 3 x} = lim_{x \to 0} \frac{\frac{d}{d x} [4 \ln (2 x + 1) + x^{3}]}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [4 \ln (2 x + 1) + x^{3}]}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.2

By the Sum Rule, the derivative of $4 \ln (2 x + 1) + x^{3}$ with respect to $x$ is $\frac{d}{d x} [4 \ln (2 x + 1)] + \frac{d}{d x} [x^{3}]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [4 \ln (2 x + 1)] + \frac{d}{d x} [x^{3}]}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.3

Evaluate $\frac{d}{d x} [4 \ln (2 x + 1)]$ .

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Step 3.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 \ln (2 x + 1)$ with respect to $x$ is $4 \frac{d}{d x} [\ln (2 x + 1)]$ .

$lim_{x \to 0} \frac{4 \frac{d}{d x} [\ln (2 x + 1)] + \frac{d}{d x} [x^{3}]}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 2 x + 1$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u$ as $2 x + 1$ .

$lim_{x \to 0} \frac{4 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [2 x + 1]) + \frac{d}{d x} [x^{3}]}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.3.2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$lim_{x \to 0} \frac{4 (\frac{1}{u} \frac{d}{d x} [2 x + 1]) + \frac{d}{d x} [x^{3}]}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.3.2.3

Replace all occurrences of $u$ with $2 x + 1$ .

$lim_{x \to 0} \frac{4 (\frac{1}{2 x + 1} \frac{d}{d x} [2 x + 1]) + \frac{d}{d x} [x^{3}]}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.3.3

By the Sum Rule, the derivative of $2 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

$lim_{x \to 0} \frac{4 (\frac{1}{2 x + 1} (\frac{d}{d x} [2 x] + \frac{d}{d x} [1])) + \frac{d}{d x} [x^{3}]}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.3.4

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

$lim_{x \to 0} \frac{4 (\frac{1}{2 x + 1} (2 \frac{d}{d x} [x] + \frac{d}{d x} [1])) + \frac{d}{d x} [x^{3}]}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.3.5

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

$lim_{x \to 0} \frac{4 (\frac{1}{2 x + 1} (2 \cdot 1 + \frac{d}{d x} [1])) + \frac{d}{d x} [x^{3}]}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.3.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{4 (\frac{1}{2 x + 1} (2 \cdot 1 + 0)) + \frac{d}{d x} [x^{3}]}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.3.7

Multiply $2$ by $1$ .

$lim_{x \to 0} \frac{4 (\frac{1}{2 x + 1} (2 + 0)) + \frac{d}{d x} [x^{3}]}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.3.8

Add $2$ and $0$ .

$lim_{x \to 0} \frac{4 (\frac{1}{2 x + 1} \cdot 2) + \frac{d}{d x} [x^{3}]}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.3.9

Combine $\frac{1}{2 x + 1}$ and $2$ .

$lim_{x \to 0} \frac{4 \frac{2}{2 x + 1} + \frac{d}{d x} [x^{3}]}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.3.10

Combine $4$ and $\frac{2}{2 x + 1}$ .

$lim_{x \to 0} \frac{\frac{4 \cdot 2}{2 x + 1} + \frac{d}{d x} [x^{3}]}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.3.11

Multiply $4$ by $2$ .

$lim_{x \to 0} \frac{\frac{8}{2 x + 1} + \frac{d}{d x} [x^{3}]}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.4

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

$lim_{x \to 0} \frac{\frac{8}{2 x + 1} + 3 x^{2}}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.5

Simplify.

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Step 3.5.1

Combine terms.

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Step 3.5.1.1

To write $3 x^{2}$ as a fraction with a common denominator, multiply by $\frac{2 x + 1}{2 x + 1}$ .

$lim_{x \to 0} \frac{\frac{8}{2 x + 1} + \frac{3 x^{2} (2 x + 1)}{2 x + 1}}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.5.1.2

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{8 + 3 x^{2} (2 x + 1)}{2 x + 1}}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.5.2

Reorder terms.

$lim_{x \to 0} \frac{\frac{3 x^{2} (2 x + 1) + 8}{2 x + 1}}{\frac{d}{d x} [5 x^{2} + 3 x]}$

Step 3.6

By the Sum Rule, the derivative of $5 x^{2} + 3 x$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [3 x]$ .

$lim_{x \to 0} \frac{\frac{3 x^{2} (2 x + 1) + 8}{2 x + 1}}{\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [3 x]}$

Step 3.7

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

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Step 3.7.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to 0} \frac{\frac{3 x^{2} (2 x + 1) + 8}{2 x + 1}}{5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x]}$

Step 3.7.2

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

$lim_{x \to 0} \frac{\frac{3 x^{2} (2 x + 1) + 8}{2 x + 1}}{5 (2 x) + \frac{d}{d x} [3 x]}$

Step 3.7.3

Multiply $2$ by $5$ .

$lim_{x \to 0} \frac{\frac{3 x^{2} (2 x + 1) + 8}{2 x + 1}}{10 x + \frac{d}{d x} [3 x]}$

Step 3.8

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

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Step 3.8.1

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

$lim_{x \to 0} \frac{\frac{3 x^{2} (2 x + 1) + 8}{2 x + 1}}{10 x + 3 \frac{d}{d x} [x]}$

Step 3.8.2

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

$lim_{x \to 0} \frac{\frac{3 x^{2} (2 x + 1) + 8}{2 x + 1}}{10 x + 3 \cdot 1}$

Step 3.8.3

Multiply $3$ by $1$ .

$lim_{x \to 0} \frac{\frac{3 x^{2} (2 x + 1) + 8}{2 x + 1}}{10 x + 3}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \frac{3 x^{2} (2 x + 1) + 8}{2 x + 1} \cdot \frac{1}{10 x + 3}$

Step 5

Multiply $\frac{3 x^{2} (2 x + 1) + 8}{2 x + 1}$ by $\frac{1}{10 x + 3}$ .

$lim_{x \to 0} \frac{3 x^{2} (2 x + 1) + 8}{(2 x + 1) (10 x + 3)}$

Step 6

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

$\frac{lim_{x \to 0} 3 x^{2} (2 x + 1) + 8}{lim_{x \to 0} (2 x + 1) (10 x + 3)}$

Step 7

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

$\frac{lim_{x \to 0} 3 x^{2} (2 x + 1) + lim_{x \to 0} 8}{lim_{x \to 0} (2 x + 1) (10 x + 3)}$

Step 8

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

$\frac{3 lim_{x \to 0} x^{2} (2 x + 1) + lim_{x \to 0} 8}{lim_{x \to 0} (2 x + 1) (10 x + 3)}$

Step 9

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

$\frac{3 lim_{x \to 0} x^{2} \cdot lim_{x \to 0} 2 x + 1 + lim_{x \to 0} 8}{lim_{x \to 0} (2 x + 1) (10 x + 3)}$

Step 10

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

$\frac{3 {(lim_{x \to 0} x)}^{2} \cdot lim_{x \to 0} 2 x + 1 + lim_{x \to 0} 8}{lim_{x \to 0} (2 x + 1) (10 x + 3)}$

Step 11

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

$\frac{3 {(lim_{x \to 0} x)}^{2} \cdot (lim_{x \to 0} 2 x + lim_{x \to 0} 1) + lim_{x \to 0} 8}{lim_{x \to 0} (2 x + 1) (10 x + 3)}$

Step 12

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

$\frac{3 {(lim_{x \to 0} x)}^{2} \cdot (2 lim_{x \to 0} x + lim_{x \to 0} 1) + lim_{x \to 0} 8}{lim_{x \to 0} (2 x + 1) (10 x + 3)}$

Step 13

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

$\frac{3 {(lim_{x \to 0} x)}^{2} \cdot (2 lim_{x \to 0} x + 1) + lim_{x \to 0} 8}{lim_{x \to 0} (2 x + 1) (10 x + 3)}$

Step 14

Evaluate the limit of $8$ which is constant as $x$ approaches $0$ .

$\frac{3 {(lim_{x \to 0} x)}^{2} \cdot (2 lim_{x \to 0} x + 1) + 8}{lim_{x \to 0} (2 x + 1) (10 x + 3)}$

Step 15

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

$\frac{3 {(lim_{x \to 0} x)}^{2} \cdot (2 lim_{x \to 0} x + 1) + 8}{lim_{x \to 0} 2 x + 1 \cdot lim_{x \to 0} 10 x + 3}$

Step 16

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

$\frac{3 {(lim_{x \to 0} x)}^{2} \cdot (2 lim_{x \to 0} x + 1) + 8}{(lim_{x \to 0} 2 x + lim_{x \to 0} 1) \cdot lim_{x \to 0} 10 x + 3}$

Step 17

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

$\frac{3 {(lim_{x \to 0} x)}^{2} \cdot (2 lim_{x \to 0} x + 1) + 8}{(2 lim_{x \to 0} x + lim_{x \to 0} 1) \cdot lim_{x \to 0} 10 x + 3}$

Step 18

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

$\frac{3 {(lim_{x \to 0} x)}^{2} \cdot (2 lim_{x \to 0} x + 1) + 8}{(2 lim_{x \to 0} x + 1) \cdot lim_{x \to 0} 10 x + 3}$

Step 19

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

$\frac{3 {(lim_{x \to 0} x)}^{2} \cdot (2 lim_{x \to 0} x + 1) + 8}{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} 10 x + lim_{x \to 0} 3)}$

Step 20

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 {(lim_{x \to 0} x)}^{2} \cdot (2 lim_{x \to 0} x + 1) + 8}{(2 lim_{x \to 0} x + 1) \cdot (10 lim_{x \to 0} x + lim_{x \to 0} 3)}$

Step 21

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

$\frac{3 {(lim_{x \to 0} x)}^{2} \cdot (2 lim_{x \to 0} x + 1) + 8}{(2 lim_{x \to 0} x + 1) \cdot (10 lim_{x \to 0} x + 3)}$

Step 22

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

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Step 22.1

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

$\frac{3 \cdot 0^{2} \cdot (2 lim_{x \to 0} x + 1) + 8}{(2 lim_{x \to 0} x + 1) \cdot (10 lim_{x \to 0} x + 3)}$

Step 22.2

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

$\frac{3 \cdot 0^{2} \cdot (2 \cdot 0 + 1) + 8}{(2 lim_{x \to 0} x + 1) \cdot (10 lim_{x \to 0} x + 3)}$

Step 22.3

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

$\frac{3 \cdot 0^{2} \cdot (2 \cdot 0 + 1) + 8}{(2 \cdot 0 + 1) \cdot (10 lim_{x \to 0} x + 3)}$

Step 22.4

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

$\frac{3 \cdot 0^{2} \cdot (2 \cdot 0 + 1) + 8}{(2 \cdot 0 + 1) \cdot (10 \cdot 0 + 3)}$

Step 23

Simplify the answer.

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Step 23.1

Simplify the numerator.

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Step 23.1.1

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

$\frac{3 \cdot 0 \cdot (2 \cdot 0 + 1) + 8}{(2 \cdot 0 + 1) \cdot (10 \cdot 0 + 3)}$

Step 23.1.2

Multiply $3$ by $0$ .

$\frac{0 \cdot (2 \cdot 0 + 1) + 8}{(2 \cdot 0 + 1) \cdot (10 \cdot 0 + 3)}$

Step 23.1.3

Multiply $2$ by $0$ .

$\frac{0 \cdot (0 + 1) + 8}{(2 \cdot 0 + 1) \cdot (10 \cdot 0 + 3)}$

Step 23.1.4

Add $0$ and $1$ .

$\frac{0 \cdot 1 + 8}{(2 \cdot 0 + 1) \cdot (10 \cdot 0 + 3)}$

Step 23.1.5

Multiply $0$ by $1$ .

$\frac{0 + 8}{(2 \cdot 0 + 1) \cdot (10 \cdot 0 + 3)}$

Step 23.1.6

Add $0$ and $8$ .

$\frac{8}{(2 \cdot 0 + 1) \cdot (10 \cdot 0 + 3)}$

Step 23.2

Simplify the denominator.

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Step 23.2.1

Multiply $2$ by $0$ .

$\frac{8}{(0 + 1) (10 \cdot 0 + 3)}$

Step 23.2.2

Add $0$ and $1$ .

$\frac{8}{1 (10 \cdot 0 + 3)}$

Step 23.2.3

Multiply $10 \cdot 0 + 3$ by $1$ .

$\frac{8}{10 \cdot 0 + 3}$

Step 23.2.4

Multiply $10$ by $0$ .

$\frac{8}{0 + 3}$

Step 23.2.5

Add $0$ and $3$ .

$\frac{8}{3}$