Calculus Examples

$lim_{x \to 4} \frac{{(2 x - 5)}^{5} - 243}{{(5 x - 12)}^{3} - {(x + 4)}^{3}}$

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.

$\frac{lim_{x \to 4} {(2 x - 5)}^{5} - 243}{lim_{x \to 4} {(5 x - 12)}^{3} - {(x + 4)}^{3}}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} {(2 x - 5)}^{5} - lim_{x \to 4} 243}{lim_{x \to 4} {(5 x - 12)}^{3} - {(x + 4)}^{3}}$

Step 1.1.2.1.2

Move the exponent $5$ from ${(2 x - 5)}^{5}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 4} 2 x - 5)}^{5} - lim_{x \to 4} 243}{lim_{x \to 4} {(5 x - 12)}^{3} - {(x + 4)}^{3}}$

Step 1.1.2.1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{{(lim_{x \to 4} 2 x - lim_{x \to 4} 5)}^{5} - lim_{x \to 4} 243}{lim_{x \to 4} {(5 x - 12)}^{3} - {(x + 4)}^{3}}$

Step 1.1.2.1.4

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

$\frac{{(2 lim_{x \to 4} x - lim_{x \to 4} 5)}^{5} - lim_{x \to 4} 243}{lim_{x \to 4} {(5 x - 12)}^{3} - {(x + 4)}^{3}}$

Step 1.1.2.1.5

Evaluate the limit of $5$ which is constant as $x$ approaches $4$ .

$\frac{{(2 lim_{x \to 4} x - 1 \cdot 5)}^{5} - lim_{x \to 4} 243}{lim_{x \to 4} {(5 x - 12)}^{3} - {(x + 4)}^{3}}$

Step 1.1.2.1.6

Evaluate the limit of $243$ which is constant as $x$ approaches $4$ .

$\frac{{(2 lim_{x \to 4} x - 1 \cdot 5)}^{5} - 1 \cdot 243}{lim_{x \to 4} {(5 x - 12)}^{3} - {(x + 4)}^{3}}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{{(2 \cdot 4 - 1 \cdot 5)}^{5} - 1 \cdot 243}{lim_{x \to 4} {(5 x - 12)}^{3} - {(x + 4)}^{3}}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Simplify each term.

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Step 1.1.2.3.1.1.1

Multiply $2$ by $4$ .

$\frac{{(8 - 1 \cdot 5)}^{5} - 1 \cdot 243}{lim_{x \to 4} {(5 x - 12)}^{3} - {(x + 4)}^{3}}$

Step 1.1.2.3.1.1.2

Multiply $- 1$ by $5$ .

$\frac{{(8 - 5)}^{5} - 1 \cdot 243}{lim_{x \to 4} {(5 x - 12)}^{3} - {(x + 4)}^{3}}$

Step 1.1.2.3.1.2

Subtract $5$ from $8$ .

$\frac{3^{5} - 1 \cdot 243}{lim_{x \to 4} {(5 x - 12)}^{3} - {(x + 4)}^{3}}$

Step 1.1.2.3.1.3

Raise $3$ to the power of $5$ .

$\frac{243 - 1 \cdot 243}{lim_{x \to 4} {(5 x - 12)}^{3} - {(x + 4)}^{3}}$

Step 1.1.2.3.1.4

Multiply $- 1$ by $243$ .

$\frac{243 - 243}{lim_{x \to 4} {(5 x - 12)}^{3} - {(x + 4)}^{3}}$

Step 1.1.2.3.2

Subtract $243$ from $243$ .

$\frac{0}{lim_{x \to 4} {(5 x - 12)}^{3} - {(x + 4)}^{3}}$

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 $4$ .

$\frac{0}{lim_{x \to 4} {(5 x - 12)}^{3} - lim_{x \to 4} {(x + 4)}^{3}}$

Step 1.1.3.2

Move the exponent $3$ from ${(5 x - 12)}^{3}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{x \to 4} 5 x - 12)}^{3} - lim_{x \to 4} {(x + 4)}^{3}}$

Step 1.1.3.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{0}{{(lim_{x \to 4} 5 x - lim_{x \to 4} 12)}^{3} - lim_{x \to 4} {(x + 4)}^{3}}$

Step 1.1.3.4

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

$\frac{0}{{(5 lim_{x \to 4} x - lim_{x \to 4} 12)}^{3} - lim_{x \to 4} {(x + 4)}^{3}}$

Step 1.1.3.5

Evaluate the limit of $12$ which is constant as $x$ approaches $4$ .

$\frac{0}{{(5 lim_{x \to 4} x - 1 \cdot 12)}^{3} - lim_{x \to 4} {(x + 4)}^{3}}$

Step 1.1.3.6

Move the exponent $3$ from ${(x + 4)}^{3}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(5 lim_{x \to 4} x - 1 \cdot 12)}^{3} - {(lim_{x \to 4} x + 4)}^{3}}$

Step 1.1.3.7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{0}{{(5 lim_{x \to 4} x - 1 \cdot 12)}^{3} - {(lim_{x \to 4} x + lim_{x \to 4} 4)}^{3}}$

Step 1.1.3.8

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

$\frac{0}{{(5 lim_{x \to 4} x - 1 \cdot 12)}^{3} - {(lim_{x \to 4} x + 4)}^{3}}$

Step 1.1.3.9

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 1.1.3.9.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{0}{{(5 \cdot 4 - 1 \cdot 12)}^{3} - {(lim_{x \to 4} x + 4)}^{3}}$

Step 1.1.3.9.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{0}{{(5 \cdot 4 - 1 \cdot 12)}^{3} - {(4 + 4)}^{3}}$

Step 1.1.3.10

Simplify the answer.

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Step 1.1.3.10.1

Simplify each term.

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Step 1.1.3.10.1.1

Simplify each term.

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Step 1.1.3.10.1.1.1

Multiply $5$ by $4$ .

$\frac{0}{{(20 - 1 \cdot 12)}^{3} - {(4 + 4)}^{3}}$

Step 1.1.3.10.1.1.2

Multiply $- 1$ by $12$ .

$\frac{0}{{(20 - 12)}^{3} - {(4 + 4)}^{3}}$

Step 1.1.3.10.1.2

Subtract $12$ from $20$ .

$\frac{0}{8^{3} - {(4 + 4)}^{3}}$

Step 1.1.3.10.1.3

Raise $8$ to the power of $3$ .

$\frac{0}{512 - {(4 + 4)}^{3}}$

Step 1.1.3.10.1.4

Add $4$ and $4$ .

$\frac{0}{512 - 8^{3}}$

Step 1.1.3.10.1.5

Raise $8$ to the power of $3$ .

$\frac{0}{512 - 1 \cdot 512}$

Step 1.1.3.10.1.6

Multiply $- 1$ by $512$ .

$\frac{0}{512 - 512}$

Step 1.1.3.10.2

Subtract $512$ from $512$ .

$\frac{0}{0}$

Step 1.1.3.10.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.11

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.

$lim_{x \to 4} \frac{{(2 x - 5)}^{5} - 243}{{(5 x - 12)}^{3} - {(x + 4)}^{3}} = lim_{x \to 4} \frac{\frac{d}{d x} [{(2 x - 5)}^{5} - 243]}{\frac{d}{d x} [{(5 x - 12)}^{3} - {(x + 4)}^{3}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 4} \frac{\frac{d}{d x} [{(2 x - 5)}^{5} - 243]}{\frac{d}{d x} [{(5 x - 12)}^{3} - {(x + 4)}^{3}]}$

Step 1.3.2

By the Sum Rule, the derivative of ${(2 x - 5)}^{5} - 243$ with respect to $x$ is $\frac{d}{d x} [{(2 x - 5)}^{5}] + \frac{d}{d x} [- 243]$ .

$lim_{x \to 4} \frac{\frac{d}{d x} [{(2 x - 5)}^{5}] + \frac{d}{d x} [- 243]}{\frac{d}{d x} [{(5 x - 12)}^{3} - {(x + 4)}^{3}]}$

Step 1.3.3

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

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Step 1.3.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = 2 x - 5$ .

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Step 1.3.3.1.1

To apply the Chain Rule, set $u$ as $2 x - 5$ .

$lim_{x \to 4} \frac{\frac{d}{d u} [u^{5}] \frac{d}{d x} [2 x - 5] + \frac{d}{d x} [- 243]}{\frac{d}{d x} [{(5 x - 12)}^{3} - {(x + 4)}^{3}]}$

Step 1.3.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$lim_{x \to 4} \frac{5 u^{4} \frac{d}{d x} [2 x - 5] + \frac{d}{d x} [- 243]}{\frac{d}{d x} [{(5 x - 12)}^{3} - {(x + 4)}^{3}]}$

Step 1.3.3.1.3

Replace all occurrences of $u$ with $2 x - 5$ .

$lim_{x \to 4} \frac{5 {(2 x - 5)}^{4} \frac{d}{d x} [2 x - 5] + \frac{d}{d x} [- 243]}{\frac{d}{d x} [{(5 x - 12)}^{3} - {(x + 4)}^{3}]}$

Step 1.3.3.2

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

$lim_{x \to 4} \frac{5 {(2 x - 5)}^{4} (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 5]) + \frac{d}{d x} [- 243]}{\frac{d}{d x} [{(5 x - 12)}^{3} - {(x + 4)}^{3}]}$

Step 1.3.3.3

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

$lim_{x \to 4} \frac{5 {(2 x - 5)}^{4} (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 5]) + \frac{d}{d x} [- 243]}{\frac{d}{d x} [{(5 x - 12)}^{3} - {(x + 4)}^{3}]}$

Step 1.3.3.4

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

$lim_{x \to 4} \frac{5 {(2 x - 5)}^{4} (2 \cdot 1 + \frac{d}{d x} [- 5]) + \frac{d}{d x} [- 243]}{\frac{d}{d x} [{(5 x - 12)}^{3} - {(x + 4)}^{3}]}$

Step 1.3.3.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{5 {(2 x - 5)}^{4} (2 \cdot 1 + 0) + \frac{d}{d x} [- 243]}{\frac{d}{d x} [{(5 x - 12)}^{3} - {(x + 4)}^{3}]}$

Step 1.3.3.6

Multiply $2$ by $1$ .

$lim_{x \to 4} \frac{5 {(2 x - 5)}^{4} (2 + 0) + \frac{d}{d x} [- 243]}{\frac{d}{d x} [{(5 x - 12)}^{3} - {(x + 4)}^{3}]}$

Step 1.3.3.7

Add $2$ and $0$ .

$lim_{x \to 4} \frac{5 {(2 x - 5)}^{4} \cdot 2 + \frac{d}{d x} [- 243]}{\frac{d}{d x} [{(5 x - 12)}^{3} - {(x + 4)}^{3}]}$

Step 1.3.3.8

Multiply $2$ by $5$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4} + \frac{d}{d x} [- 243]}{\frac{d}{d x} [{(5 x - 12)}^{3} - {(x + 4)}^{3}]}$

Step 1.3.4

Since $- 243$ is constant with respect to $x$ , the derivative of $- 243$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4} + 0}{\frac{d}{d x} [{(5 x - 12)}^{3} - {(x + 4)}^{3}]}$

Step 1.3.5

Add $10 {(2 x - 5)}^{4}$ and $0$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{\frac{d}{d x} [{(5 x - 12)}^{3} - {(x + 4)}^{3}]}$

Step 1.3.6

By the Sum Rule, the derivative of ${(5 x - 12)}^{3} - {(x + 4)}^{3}$ with respect to $x$ is $\frac{d}{d x} [{(5 x - 12)}^{3}] + \frac{d}{d x} [- {(x + 4)}^{3}]$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{\frac{d}{d x} [{(5 x - 12)}^{3}] + \frac{d}{d x} [- {(x + 4)}^{3}]}$

Step 1.3.7

Evaluate $\frac{d}{d x} [{(5 x - 12)}^{3}]$ .

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Step 1.3.7.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 5 x - 12$ .

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Step 1.3.7.1.1

To apply the Chain Rule, set $u_{1}$ as $5 x - 12$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [5 x - 12] + \frac{d}{d x} [- {(x + 4)}^{3}]}$

Step 1.3.7.1.2

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

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{3 {u_{1}}^{2} \frac{d}{d x} [5 x - 12] + \frac{d}{d x} [- {(x + 4)}^{3}]}$

Step 1.3.7.1.3

Replace all occurrences of $u_{1}$ with $5 x - 12$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{3 {(5 x - 12)}^{2} \frac{d}{d x} [5 x - 12] + \frac{d}{d x} [- {(x + 4)}^{3}]}$

Step 1.3.7.2

By the Sum Rule, the derivative of $5 x - 12$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 12]$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{3 {(5 x - 12)}^{2} (\frac{d}{d x} [5 x] + \frac{d}{d x} [- 12]) + \frac{d}{d x} [- {(x + 4)}^{3}]}$

Step 1.3.7.3

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

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{3 {(5 x - 12)}^{2} (5 \frac{d}{d x} [x] + \frac{d}{d x} [- 12]) + \frac{d}{d x} [- {(x + 4)}^{3}]}$

Step 1.3.7.4

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

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{3 {(5 x - 12)}^{2} (5 \cdot 1 + \frac{d}{d x} [- 12]) + \frac{d}{d x} [- {(x + 4)}^{3}]}$

Step 1.3.7.5

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{3 {(5 x - 12)}^{2} (5 \cdot 1 + 0) + \frac{d}{d x} [- {(x + 4)}^{3}]}$

Step 1.3.7.6

Multiply $5$ by $1$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{3 {(5 x - 12)}^{2} (5 + 0) + \frac{d}{d x} [- {(x + 4)}^{3}]}$

Step 1.3.7.7

Add $5$ and $0$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{3 {(5 x - 12)}^{2} \cdot 5 + \frac{d}{d x} [- {(x + 4)}^{3}]}$

Step 1.3.7.8

Multiply $5$ by $3$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 {(5 x - 12)}^{2} + \frac{d}{d x} [- {(x + 4)}^{3}]}$

Step 1.3.8

Evaluate $\frac{d}{d x} [- {(x + 4)}^{3}]$ .

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Step 1.3.8.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(x + 4)}^{3}$ with respect to $x$ is $- \frac{d}{d x} [{(x + 4)}^{3}]$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 {(5 x - 12)}^{2} - \frac{d}{d x} [{(x + 4)}^{3}]}$

Step 1.3.8.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x + 4$ .

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Step 1.3.8.2.1

To apply the Chain Rule, set $u_{2}$ as $x + 4$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 {(5 x - 12)}^{2} - (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d x} [x + 4])}$

Step 1.3.8.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 3$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 {(5 x - 12)}^{2} - (3 {u_{2}}^{2} \frac{d}{d x} [x + 4])}$

Step 1.3.8.2.3

Replace all occurrences of $u_{2}$ with $x + 4$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 {(5 x - 12)}^{2} - (3 {(x + 4)}^{2} \frac{d}{d x} [x + 4])}$

Step 1.3.8.3

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

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 {(5 x - 12)}^{2} - (3 {(x + 4)}^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [4]))}$

Step 1.3.8.4

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

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 {(5 x - 12)}^{2} - (3 {(x + 4)}^{2} (1 + \frac{d}{d x} [4]))}$

Step 1.3.8.5

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 {(5 x - 12)}^{2} - (3 {(x + 4)}^{2} (1 + 0))}$

Step 1.3.8.6

Add $1$ and $0$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 {(5 x - 12)}^{2} - (3 {(x + 4)}^{2} \cdot 1)}$

Step 1.3.8.7

Multiply $3$ by $1$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 {(5 x - 12)}^{2} - (3 {(x + 4)}^{2})}$

Step 1.3.8.8

Multiply $3$ by $- 1$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 {(5 x - 12)}^{2} - 3 {(x + 4)}^{2}}$

Step 1.3.9

Simplify.

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Step 1.3.9.1

Simplify each term.

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Step 1.3.9.1.1

Rewrite ${(5 x - 12)}^{2}$ as $(5 x - 12) (5 x - 12)$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 ((5 x - 12) (5 x - 12)) - 3 {(x + 4)}^{2}}$

Step 1.3.9.1.2

Expand $(5 x - 12) (5 x - 12)$ using the FOIL Method.

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Step 1.3.9.1.2.1

Apply the distributive property.

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 (5 x (5 x - 12) - 12 (5 x - 12)) - 3 {(x + 4)}^{2}}$

Step 1.3.9.1.2.2

Apply the distributive property.

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 (5 x (5 x) + 5 x \cdot - 12 - 12 (5 x - 12)) - 3 {(x + 4)}^{2}}$

Step 1.3.9.1.2.3

Apply the distributive property.

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 (5 x (5 x) + 5 x \cdot - 12 - 12 (5 x) - 12 \cdot - 12) - 3 {(x + 4)}^{2}}$

Step 1.3.9.1.3

Simplify and combine like terms.

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Step 1.3.9.1.3.1

Simplify each term.

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Step 1.3.9.1.3.1.1

Rewrite using the commutative property of multiplication.

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 (5 \cdot 5 x \cdot x + 5 x \cdot - 12 - 12 (5 x) - 12 \cdot - 12) - 3 {(x + 4)}^{2}}$

Step 1.3.9.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.9.1.3.1.2.1

Move $x$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 (5 \cdot 5 (x \cdot x) + 5 x \cdot - 12 - 12 (5 x) - 12 \cdot - 12) - 3 {(x + 4)}^{2}}$

Step 1.3.9.1.3.1.2.2

Multiply $x$ by $x$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 (5 \cdot 5 x^{2} + 5 x \cdot - 12 - 12 (5 x) - 12 \cdot - 12) - 3 {(x + 4)}^{2}}$

Step 1.3.9.1.3.1.3

Multiply $5$ by $5$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 (25 x^{2} + 5 x \cdot - 12 - 12 (5 x) - 12 \cdot - 12) - 3 {(x + 4)}^{2}}$

Step 1.3.9.1.3.1.4

Multiply $- 12$ by $5$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 (25 x^{2} - 60 x - 12 (5 x) - 12 \cdot - 12) - 3 {(x + 4)}^{2}}$

Step 1.3.9.1.3.1.5

Multiply $5$ by $- 12$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 (25 x^{2} - 60 x - 60 x - 12 \cdot - 12) - 3 {(x + 4)}^{2}}$

Step 1.3.9.1.3.1.6

Multiply $- 12$ by $- 12$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 (25 x^{2} - 60 x - 60 x + 144) - 3 {(x + 4)}^{2}}$

Step 1.3.9.1.3.2

Subtract $60 x$ from $- 60 x$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 (25 x^{2} - 120 x + 144) - 3 {(x + 4)}^{2}}$

Step 1.3.9.1.4

Apply the distributive property.

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{15 (25 x^{2}) + 15 (- 120 x) + 15 \cdot 144 - 3 {(x + 4)}^{2}}$

Step 1.3.9.1.5

Simplify.

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Step 1.3.9.1.5.1

Multiply $25$ by $15$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{375 x^{2} + 15 (- 120 x) + 15 \cdot 144 - 3 {(x + 4)}^{2}}$

Step 1.3.9.1.5.2

Multiply $- 120$ by $15$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{375 x^{2} - 1800 x + 15 \cdot 144 - 3 {(x + 4)}^{2}}$

Step 1.3.9.1.5.3

Multiply $15$ by $144$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{375 x^{2} - 1800 x + 2160 - 3 {(x + 4)}^{2}}$

Step 1.3.9.1.6

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{375 x^{2} - 1800 x + 2160 - 3 ((x + 4) (x + 4))}$

Step 1.3.9.1.7

Expand $(x + 4) (x + 4)$ using the FOIL Method.

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Step 1.3.9.1.7.1

Apply the distributive property.

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{375 x^{2} - 1800 x + 2160 - 3 (x (x + 4) + 4 (x + 4))}$

Step 1.3.9.1.7.2

Apply the distributive property.

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{375 x^{2} - 1800 x + 2160 - 3 (x \cdot x + x \cdot 4 + 4 (x + 4))}$

Step 1.3.9.1.7.3

Apply the distributive property.

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{375 x^{2} - 1800 x + 2160 - 3 (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4)}$

Step 1.3.9.1.8

Simplify and combine like terms.

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Step 1.3.9.1.8.1

Simplify each term.

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Step 1.3.9.1.8.1.1

Multiply $x$ by $x$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{375 x^{2} - 1800 x + 2160 - 3 (x^{2} + x \cdot 4 + 4 x + 4 \cdot 4)}$

Step 1.3.9.1.8.1.2

Move $4$ to the left of $x$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{375 x^{2} - 1800 x + 2160 - 3 (x^{2} + 4 \cdot x + 4 x + 4 \cdot 4)}$

Step 1.3.9.1.8.1.3

Multiply $4$ by $4$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{375 x^{2} - 1800 x + 2160 - 3 (x^{2} + 4 x + 4 x + 16)}$

Step 1.3.9.1.8.2

Add $4 x$ and $4 x$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{375 x^{2} - 1800 x + 2160 - 3 (x^{2} + 8 x + 16)}$

Step 1.3.9.1.9

Apply the distributive property.

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{375 x^{2} - 1800 x + 2160 - 3 x^{2} - 3 (8 x) - 3 \cdot 16}$

Step 1.3.9.1.10

Simplify.

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Step 1.3.9.1.10.1

Multiply $8$ by $- 3$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{375 x^{2} - 1800 x + 2160 - 3 x^{2} - 24 x - 3 \cdot 16}$

Step 1.3.9.1.10.2

Multiply $- 3$ by $16$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{375 x^{2} - 1800 x + 2160 - 3 x^{2} - 24 x - 48}$

Step 1.3.9.2

Subtract $3 x^{2}$ from $375 x^{2}$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{372 x^{2} - 1800 x + 2160 - 24 x - 48}$

Step 1.3.9.3

Subtract $24 x$ from $- 1800 x$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{372 x^{2} - 1824 x + 2160 - 48}$

Step 1.3.9.4

Subtract $48$ from $2160$ .

$lim_{x \to 4} \frac{10 {(2 x - 5)}^{4}}{372 x^{2} - 1824 x + 2112}$

Step 1.4

Cancel the common factor of $10$ and $372 x^{2} - 1824 x + 2112$ .

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Step 1.4.1

Factor $2$ out of $10 {(2 x - 5)}^{4}$ .

$lim_{x \to 4} \frac{2 (5 {(2 x - 5)}^{4})}{372 x^{2} - 1824 x + 2112}$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $2$ out of $372 x^{2}$ .

$lim_{x \to 4} \frac{2 (5 {(2 x - 5)}^{4})}{2 (186 x^{2}) - 1824 x + 2112}$

Step 1.4.2.2

Factor $2$ out of $- 1824 x$ .

$lim_{x \to 4} \frac{2 (5 {(2 x - 5)}^{4})}{2 (186 x^{2}) + 2 (- 912 x) + 2112}$

Step 1.4.2.3

Factor $2$ out of $2 (186 x^{2}) + 2 (- 912 x)$ .

$lim_{x \to 4} \frac{2 (5 {(2 x - 5)}^{4})}{2 (186 x^{2} - 912 x) + 2112}$

Step 1.4.2.4

Factor $2$ out of $2112$ .

$lim_{x \to 4} \frac{2 (5 {(2 x - 5)}^{4})}{2 (186 x^{2} - 912 x) + 2 (1056)}$

Step 1.4.2.5

Factor $2$ out of $2 (186 x^{2} - 912 x) + 2 (1056)$ .

$lim_{x \to 4} \frac{2 (5 {(2 x - 5)}^{4})}{2 (186 x^{2} - 912 x + 1056)}$

Step 1.4.2.6

Cancel the common factor.

$lim_{x \to 4} \frac{2 (5 {(2 x - 5)}^{4})}{2 (186 x^{2} - 912 x + 1056)}$

Step 1.4.2.7

Rewrite the expression.

$lim_{x \to 4} \frac{5 {(2 x - 5)}^{4}}{186 x^{2} - 912 x + 1056}$

Step 2

Evaluate the limit.

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Step 2.1

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

$5 lim_{x \to 4} \frac{{(2 x - 5)}^{4}}{186 x^{2} - 912 x + 1056}$

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

$5 \frac{lim_{x \to 4} {(2 x - 5)}^{4}}{lim_{x \to 4} 186 x^{2} - 912 x + 1056}$

Step 2.3

Move the exponent $4$ from ${(2 x - 5)}^{4}$ outside the limit using the Limits Power Rule.

$5 \frac{{(lim_{x \to 4} 2 x - 5)}^{4}}{lim_{x \to 4} 186 x^{2} - 912 x + 1056}$

Step 2.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$5 \frac{{(lim_{x \to 4} 2 x - lim_{x \to 4} 5)}^{4}}{lim_{x \to 4} 186 x^{2} - 912 x + 1056}$

Step 2.5

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

$5 \frac{{(2 lim_{x \to 4} x - lim_{x \to 4} 5)}^{4}}{lim_{x \to 4} 186 x^{2} - 912 x + 1056}$

Step 2.6

Evaluate the limit of $5$ which is constant as $x$ approaches $4$ .

$5 \frac{{(2 lim_{x \to 4} x - 1 \cdot 5)}^{4}}{lim_{x \to 4} 186 x^{2} - 912 x + 1056}$

Step 2.7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$5 \frac{{(2 lim_{x \to 4} x - 1 \cdot 5)}^{4}}{lim_{x \to 4} 186 x^{2} - lim_{x \to 4} 912 x + lim_{x \to 4} 1056}$

Step 2.8

Move the term $186$ outside of the limit because it is constant with respect to $x$ .

$5 \frac{{(2 lim_{x \to 4} x - 1 \cdot 5)}^{4}}{186 lim_{x \to 4} x^{2} - lim_{x \to 4} 912 x + lim_{x \to 4} 1056}$

Step 2.9

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

$5 \frac{{(2 lim_{x \to 4} x - 1 \cdot 5)}^{4}}{186 {(lim_{x \to 4} x)}^{2} - lim_{x \to 4} 912 x + lim_{x \to 4} 1056}$

Step 2.10

Move the term $912$ outside of the limit because it is constant with respect to $x$ .

$5 \frac{{(2 lim_{x \to 4} x - 1 \cdot 5)}^{4}}{186 {(lim_{x \to 4} x)}^{2} - 912 lim_{x \to 4} x + lim_{x \to 4} 1056}$

Step 2.11

Evaluate the limit of $1056$ which is constant as $x$ approaches $4$ .

$5 \frac{{(2 lim_{x \to 4} x - 1 \cdot 5)}^{4}}{186 {(lim_{x \to 4} x)}^{2} - 912 lim_{x \to 4} x + 1056}$

Step 3

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$5 \frac{{(2 \cdot 4 - 1 \cdot 5)}^{4}}{186 {(lim_{x \to 4} x)}^{2} - 912 lim_{x \to 4} x + 1056}$

Step 3.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$5 \frac{{(2 \cdot 4 - 1 \cdot 5)}^{4}}{186 \cdot 4^{2} - 912 lim_{x \to 4} x + 1056}$

Step 3.3

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$5 \frac{{(2 \cdot 4 - 1 \cdot 5)}^{4}}{186 \cdot 4^{2} - 912 \cdot 4 + 1056}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $2$ by $4$ .

$5 \frac{{(8 - 1 \cdot 5)}^{4}}{186 \cdot 4^{2} - 912 \cdot 4 + 1056}$

Step 4.1.2

Multiply $- 1$ by $5$ .

$5 \frac{{(8 - 5)}^{4}}{186 \cdot 4^{2} - 912 \cdot 4 + 1056}$

Step 4.1.3

Subtract $5$ from $8$ .

$5 \frac{3^{4}}{186 \cdot 4^{2} - 912 \cdot 4 + 1056}$

Step 4.1.4

Raise $3$ to the power of $4$ .

$5 \frac{81}{186 \cdot 4^{2} - 912 \cdot 4 + 1056}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Raise $4$ to the power of $2$ .

$5 \frac{81}{186 \cdot 16 - 912 \cdot 4 + 1056}$

Step 4.2.2

Multiply $186$ by $16$ .

$5 \frac{81}{2976 - 912 \cdot 4 + 1056}$

Step 4.2.3

Multiply $- 912$ by $4$ .

$5 \frac{81}{2976 - 3648 + 1056}$

Step 4.2.4

Subtract $3648$ from $2976$ .

$5 \frac{81}{- 672 + 1056}$

Step 4.2.5

Add $- 672$ and $1056$ .

$5 (\frac{81}{384})$

Step 4.3

Cancel the common factor of $81$ and $384$ .

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Step 4.3.1

Factor $3$ out of $81$ .

$5 \frac{3 (27)}{384}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $3$ out of $384$ .

$5 \frac{3 \cdot 27}{3 \cdot 128}$

Step 4.3.2.2

Cancel the common factor.

$5 \frac{3 \cdot 27}{3 \cdot 128}$

Step 4.3.2.3

Rewrite the expression.

$5 (\frac{27}{128})$

Step 4.4

Multiply $5 (\frac{27}{128})$ .

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Step 4.4.1

Combine $5$ and $\frac{27}{128}$ .

$\frac{5 \cdot 27}{128}$

Step 4.4.2

Multiply $5$ by $27$ .

$\frac{135}{128}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{135}{128}$

Decimal Form:

$1.0546875$