Calculus Examples

$lim_{x \to 4} \frac{- 5 x^{2} + 21 x - 4}{24 - 6 x}$

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.

$\frac{lim_{x \to 4} - 5 x^{2} + 21 x - 4}{lim_{x \to 4} 24 - 6 x}$

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 $4$ .

$\frac{- lim_{x \to 4} 5 x^{2} + lim_{x \to 4} 21 x - lim_{x \to 4} 4}{lim_{x \to 4} 24 - 6 x}$

Step 1.1.2.2

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

$\frac{- 5 lim_{x \to 4} x^{2} + lim_{x \to 4} 21 x - lim_{x \to 4} 4}{lim_{x \to 4} 24 - 6 x}$

Step 1.1.2.3

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

$\frac{- 5 {(lim_{x \to 4} x)}^{2} + lim_{x \to 4} 21 x - lim_{x \to 4} 4}{lim_{x \to 4} 24 - 6 x}$

Step 1.1.2.4

Move the term $21$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 5 {(lim_{x \to 4} x)}^{2} + 21 lim_{x \to 4} x - lim_{x \to 4} 4}{lim_{x \to 4} 24 - 6 x}$

Step 1.1.2.5

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

$\frac{- 5 {(lim_{x \to 4} x)}^{2} + 21 lim_{x \to 4} x - 1 \cdot 4}{lim_{x \to 4} 24 - 6 x}$

Step 1.1.2.6

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

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Step 1.1.2.6.1

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

$\frac{- 5 \cdot 4^{2} + 21 lim_{x \to 4} x - 1 \cdot 4}{lim_{x \to 4} 24 - 6 x}$

Step 1.1.2.6.2

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

$\frac{- 5 \cdot 4^{2} + 21 \cdot 4 - 1 \cdot 4}{lim_{x \to 4} 24 - 6 x}$

Step 1.1.2.7

Simplify the answer.

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Step 1.1.2.7.1

Simplify each term.

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Step 1.1.2.7.1.1

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

$\frac{- 5 \cdot 16 + 21 \cdot 4 - 1 \cdot 4}{lim_{x \to 4} 24 - 6 x}$

Step 1.1.2.7.1.2

Multiply $- 5$ by $16$ .

$\frac{- 80 + 21 \cdot 4 - 1 \cdot 4}{lim_{x \to 4} 24 - 6 x}$

Step 1.1.2.7.1.3

Multiply $21$ by $4$ .

$\frac{- 80 + 84 - 1 \cdot 4}{lim_{x \to 4} 24 - 6 x}$

Step 1.1.2.7.1.4

Multiply $- 1$ by $4$ .

$\frac{- 80 + 84 - 4}{lim_{x \to 4} 24 - 6 x}$

Step 1.1.2.7.2

Add $- 80$ and $84$ .

$\frac{4 - 4}{lim_{x \to 4} 24 - 6 x}$

Step 1.1.2.7.3

Subtract $4$ from $4$ .

$\frac{0}{lim_{x \to 4} 24 - 6 x}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

$\frac{0}{lim_{x \to 4} 24 - lim_{x \to 4} 6 x}$

Step 1.1.3.1.2

Evaluate the limit of $24$ which is constant as $x$ approaches $4$ .

$\frac{0}{24 - lim_{x \to 4} 6 x}$

Step 1.1.3.1.3

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

$\frac{0}{24 - 6 lim_{x \to 4} x}$

Step 1.1.3.2

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

$\frac{0}{24 - 6 \cdot 4}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $- 6$ by $4$ .

$\frac{0}{24 - 24}$

Step 1.1.3.3.2

Subtract $24$ from $24$ .

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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.

$lim_{x \to 4} \frac{- 5 x^{2} + 21 x - 4}{24 - 6 x} = lim_{x \to 4} \frac{\frac{d}{d x} [- 5 x^{2} + 21 x - 4]}{\frac{d}{d x} [24 - 6 x]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 4} \frac{\frac{d}{d x} [- 5 x^{2} + 21 x - 4]}{\frac{d}{d x} [24 - 6 x]}$

Step 1.3.2

By the Sum Rule, the derivative of $- 5 x^{2} + 21 x - 4$ with respect to $x$ is $\frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [21 x] + \frac{d}{d x} [- 4]$ .

$lim_{x \to 4} \frac{\frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [21 x] + \frac{d}{d x} [- 4]}{\frac{d}{d x} [24 - 6 x]}$

Step 1.3.3

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

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Step 1.3.3.1

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

$lim_{x \to 4} \frac{- 5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [21 x] + \frac{d}{d x} [- 4]}{\frac{d}{d x} [24 - 6 x]}$

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 $n = 2$ .

$lim_{x \to 4} \frac{- 5 (2 x) + \frac{d}{d x} [21 x] + \frac{d}{d x} [- 4]}{\frac{d}{d x} [24 - 6 x]}$

Step 1.3.3.3

Multiply $2$ by $- 5$ .

$lim_{x \to 4} \frac{- 10 x + \frac{d}{d x} [21 x] + \frac{d}{d x} [- 4]}{\frac{d}{d x} [24 - 6 x]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [21 x]$ .

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Step 1.3.4.1

Since $21$ is constant with respect to $x$ , the derivative of $21 x$ with respect to $x$ is $21 \frac{d}{d x} [x]$ .

$lim_{x \to 4} \frac{- 10 x + 21 \frac{d}{d x} [x] + \frac{d}{d x} [- 4]}{\frac{d}{d x} [24 - 6 x]}$

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 $n = 1$ .

$lim_{x \to 4} \frac{- 10 x + 21 \cdot 1 + \frac{d}{d x} [- 4]}{\frac{d}{d x} [24 - 6 x]}$

Step 1.3.4.3

Multiply $21$ by $1$ .

$lim_{x \to 4} \frac{- 10 x + 21 + \frac{d}{d x} [- 4]}{\frac{d}{d x} [24 - 6 x]}$

Step 1.3.5

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

$lim_{x \to 4} \frac{- 10 x + 21 + 0}{\frac{d}{d x} [24 - 6 x]}$

Step 1.3.6

Add $- 10 x + 21$ and $0$ .

$lim_{x \to 4} \frac{- 10 x + 21}{\frac{d}{d x} [24 - 6 x]}$

Step 1.3.7

By the Sum Rule, the derivative of $24 - 6 x$ with respect to $x$ is $\frac{d}{d x} [24] + \frac{d}{d x} [- 6 x]$ .

$lim_{x \to 4} \frac{- 10 x + 21}{\frac{d}{d x} [24] + \frac{d}{d x} [- 6 x]}$

Step 1.3.8

Since $24$ is constant with respect to $x$ , the derivative of $24$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{- 10 x + 21}{0 + \frac{d}{d x} [- 6 x]}$

Step 1.3.9

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

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Step 1.3.9.1

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

$lim_{x \to 4} \frac{- 10 x + 21}{0 - 6 \frac{d}{d x} [x]}$

Step 1.3.9.2

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

$lim_{x \to 4} \frac{- 10 x + 21}{0 - 6 \cdot 1}$

Step 1.3.9.3

Multiply $- 6$ by $1$ .

$lim_{x \to 4} \frac{- 10 x + 21}{0 - 6}$

Step 1.3.10

Subtract $6$ from $0$ .

$lim_{x \to 4} \frac{- 10 x + 21}{- 6}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{1}{- 6}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{- 6} lim_{x \to 4} - 10 x + 21$

Step 2.2

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

$\frac{1}{- 6} (- lim_{x \to 4} 10 x + lim_{x \to 4} 21)$

Step 2.3

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

$\frac{1}{- 6} (- 10 lim_{x \to 4} x + lim_{x \to 4} 21)$

Step 2.4

Evaluate the limit of $21$ which is constant as $x$ approaches $4$ .

$\frac{1}{- 6} (- 10 lim_{x \to 4} x + 21)$

Step 3

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

$\frac{1}{- 6} (- 10 \cdot 4 + 21)$

Step 4

Simplify the answer.

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Step 4.1

Move the negative in front of the fraction.

$- \frac{1}{6} (- 10 \cdot 4 + 21)$

Step 4.2

Multiply $- 10$ by $4$ .

$- \frac{1}{6} (- 40 + 21)$

Step 4.3

Add $- 40$ and $21$ .

$- \frac{1}{6} \cdot - 19$

Step 4.4

Multiply $- \frac{1}{6} \cdot - 19$ .

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Step 4.4.1

Multiply $- 19$ by $- 1$ .

$19 (\frac{1}{6})$

Step 4.4.2

Combine $19$ and $\frac{1}{6}$ .

$\frac{19}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{19}{6}$

Decimal Form:

$3.1\overline{6}$