Calculus Examples

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

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.

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

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 $0$ .

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

Step 1.2.2

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

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

Step 1.2.3

Move the limit inside the logarithm.

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

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

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

Step 1.2.8

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

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

Step 1.2.9

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

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Step 1.2.9.1

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

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

Step 1.2.9.2

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

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

Step 1.2.10

Simplify the answer.

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Step 1.2.10.1

Simplify each term.

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Step 1.2.10.1.1

Multiply $- 2$ by $0$ .

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

Step 1.2.10.1.2

Add $1$ and $0$ .

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

Step 1.2.10.1.3

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

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

Step 1.2.10.1.4

Multiply $4$ by $0$ .

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

Step 1.2.10.1.5

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

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

Step 1.2.10.1.6

Multiply $2$ by $0$ .

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

Step 1.2.10.2

Add $0$ and $0$ .

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

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 $0$ .

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

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 $x$ .

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

Step 1.3.5.2

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

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

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 $0$ .

$\frac{0}{5 \cdot 0 + 4 \cdot 0}$

Step 1.3.6.1.2

Multiply $5$ by $0$ .

$\frac{0}{0 + 4 \cdot 0}$

Step 1.3.6.1.3

Multiply $4$ 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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

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 $g (x) = 1 - 2 x$ .

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Step 3.3.2.1

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

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

Step 3.3.2.2

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

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

Step 3.3.2.3

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

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

Step 3.3.6

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

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

Step 3.3.7

Multiply $- 2$ by $1$ .

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

Step 3.3.8

Subtract $2$ from $0$ .

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

Step 3.3.9

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

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

Step 3.3.10

Move the negative in front of the fraction.

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

Step 3.3.11

Multiply $- 1$ by $4$ .

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

Step 3.3.12

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

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

Step 3.3.13

Multiply $- 4$ by $2$ .

$lim_{x \to 0} \frac{\frac{- 8}{1 - 2 x} + \frac{d}{d x} [2 x^{3}]}{\frac{d}{d x} [5 x^{2} + 4 x]}$

Step 3.3.14

Move the negative in front of the fraction.

$lim_{x \to 0} \frac{- \frac{8}{1 - 2 x} + \frac{d}{d x} [2 x^{3}]}{\frac{d}{d x} [5 x^{2} + 4 x]}$

Step 3.4

Evaluate $\frac{d}{d x} [2 x^{3}]$ .

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Step 3.4.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$lim_{x \to 0} \frac{- \frac{8}{1 - 2 x} + 2 \frac{d}{d x} [x^{3}]}{\frac{d}{d x} [5 x^{2} + 4 x]}$

Step 3.4.2

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

$lim_{x \to 0} \frac{- \frac{8}{1 - 2 x} + 2 (3 x^{2})}{\frac{d}{d x} [5 x^{2} + 4 x]}$

Step 3.4.3

Multiply $3$ by $2$ .

$lim_{x \to 0} \frac{- \frac{8}{1 - 2 x} + 6 x^{2}}{\frac{d}{d x} [5 x^{2} + 4 x]}$

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 $6 x^{2}$ as a fraction with a common denominator, multiply by $\frac{1 - 2 x}{1 - 2 x}$ .

$lim_{x \to 0} \frac{- \frac{8}{1 - 2 x} + \frac{6 x^{2} (1 - 2 x)}{1 - 2 x}}{\frac{d}{d x} [5 x^{2} + 4 x]}$

Step 3.5.1.2

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{- 8 + 6 x^{2} (1 - 2 x)}{1 - 2 x}}{\frac{d}{d x} [5 x^{2} + 4 x]}$

Step 3.5.2

Reorder terms.

$lim_{x \to 0} \frac{\frac{6 x^{2} (1 - 2 x) - 8}{- 2 x + 1}}{\frac{d}{d x} [5 x^{2} + 4 x]}$

Step 3.6

By the Sum Rule, the derivative of $5 x^{2} + 4 x$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [4 x]$ .

$lim_{x \to 0} \frac{\frac{6 x^{2} (1 - 2 x) - 8}{- 2 x + 1}}{\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [4 x]}$

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

$lim_{x \to 0} \frac{\frac{6 x^{2} (1 - 2 x) - 8}{- 2 x + 1}}{5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x]}$

Step 3.7.2

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

$lim_{x \to 0} \frac{\frac{6 x^{2} (1 - 2 x) - 8}{- 2 x + 1}}{5 (2 x) + \frac{d}{d x} [4 x]}$

Step 3.7.3

Multiply $2$ by $5$ .

$lim_{x \to 0} \frac{\frac{6 x^{2} (1 - 2 x) - 8}{- 2 x + 1}}{10 x + \frac{d}{d x} [4 x]}$

Step 3.8

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 3.8.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{\frac{6 x^{2} (1 - 2 x) - 8}{- 2 x + 1}}{10 x + 4 \frac{d}{d x} [x]}$

Step 3.8.2

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

$lim_{x \to 0} \frac{\frac{6 x^{2} (1 - 2 x) - 8}{- 2 x + 1}}{10 x + 4 \cdot 1}$

Step 3.8.3

Multiply $4$ by $1$ .

$lim_{x \to 0} \frac{\frac{6 x^{2} (1 - 2 x) - 8}{- 2 x + 1}}{10 x + 4}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \frac{6 x^{2} (1 - 2 x) - 8}{- 2 x + 1} \cdot \frac{1}{10 x + 4}$

Step 5

Simplify terms.

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Step 5.1

Multiply $\frac{6 x^{2} (1 - 2 x) - 8}{- 2 x + 1}$ by $\frac{1}{10 x + 4}$ .

$lim_{x \to 0} \frac{6 x^{2} (1 - 2 x) - 8}{(- 2 x + 1) (10 x + 4)}$

Step 5.2

Cancel the common factor of $6 x^{2} (1 - 2 x) - 8$ and $10 x + 4$ .

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Step 5.2.1

Factor $2$ out of $6 x^{2} (1 - 2 x)$ .

$lim_{x \to 0} \frac{2 (3 x^{2} (1 - 2 x)) - 8}{(- 2 x + 1) (10 x + 4)}$

Step 5.2.2

Factor $2$ out of $- 8$ .

$lim_{x \to 0} \frac{2 (3 x^{2} (1 - 2 x)) + 2 (- 4)}{(- 2 x + 1) (10 x + 4)}$

Step 5.2.3

Factor $2$ out of $2 (3 x^{2} (1 - 2 x)) + 2 (- 4)$ .

$lim_{x \to 0} \frac{2 (3 x^{2} (1 - 2 x) - 4)}{(- 2 x + 1) (10 x + 4)}$

Step 5.2.4

Cancel the common factors.

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Step 5.2.4.1

Factor $2$ out of $(- 2 x + 1) (10 x + 4)$ .

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

Step 5.2.4.2

Cancel the common factor.

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

Step 5.2.4.3

Rewrite the expression.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

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

Step 15

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

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

Step 16

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

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

Step 17

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

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

Step 18

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

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

Step 19

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

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

Step 20

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

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

Step 21

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

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

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 $x$ .

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

Step 22.2

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

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

Step 22.3

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

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

Step 22.4

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

$\frac{3 \cdot 0^{2} \cdot (1 - 2 \cdot 0) - 1 \cdot 4}{(- 2 \cdot 0 + 1) \cdot (5 \cdot 0 + 2)}$

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 $0$ .

$\frac{3 \cdot 0 \cdot (1 - 2 \cdot 0) - 1 \cdot 4}{(- 2 \cdot 0 + 1) \cdot (5 \cdot 0 + 2)}$

Step 23.1.2

Multiply $3$ by $0$ .

$\frac{0 \cdot (1 - 2 \cdot 0) - 1 \cdot 4}{(- 2 \cdot 0 + 1) \cdot (5 \cdot 0 + 2)}$

Step 23.1.3

Multiply $- 2$ by $0$ .

$\frac{0 \cdot (1 + 0) - 1 \cdot 4}{(- 2 \cdot 0 + 1) \cdot (5 \cdot 0 + 2)}$

Step 23.1.4

Add $1$ and $0$ .

$\frac{0 \cdot 1 - 1 \cdot 4}{(- 2 \cdot 0 + 1) \cdot (5 \cdot 0 + 2)}$

Step 23.1.5

Multiply $0$ by $1$ .

$\frac{0 - 1 \cdot 4}{(- 2 \cdot 0 + 1) \cdot (5 \cdot 0 + 2)}$

Step 23.1.6

Multiply $- 1$ by $4$ .

$\frac{0 - 4}{(- 2 \cdot 0 + 1) \cdot (5 \cdot 0 + 2)}$

Step 23.1.7

Subtract $4$ from $0$ .

$\frac{- 4}{(- 2 \cdot 0 + 1) \cdot (5 \cdot 0 + 2)}$

Step 23.2

Simplify the denominator.

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Step 23.2.1

Multiply $- 2$ by $0$ .

$\frac{- 4}{(0 + 1) (5 \cdot 0 + 2)}$

Step 23.2.2

Add $0$ and $1$ .

$\frac{- 4}{1 (5 \cdot 0 + 2)}$

Step 23.2.3

Multiply $5 \cdot 0 + 2$ by $1$ .

$\frac{- 4}{5 \cdot 0 + 2}$

Step 23.2.4

Multiply $5$ by $0$ .

$\frac{- 4}{0 + 2}$

Step 23.2.5

Add $0$ and $2$ .

$\frac{- 4}{2}$

Step 23.3

Divide $- 4$ by $2$ .

$- 2$