Calculus Examples

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

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.

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

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 $1$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

Evaluate the limit of $3 a$ which is constant as $x$ approaches $1$ .

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

Step 1.1.2.5

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

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

Step 1.1.2.6

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 1.1.2.6.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 1.1.2.6.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

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

One to any power is one.

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

Step 1.1.2.7.1.2

Multiply $- 1$ by $3$ .

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

Step 1.1.2.7.1.3

Multiply $- 3$ by $1$ .

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

Step 1.1.2.7.1.4

Multiply $- 1$ by $1$ .

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

Step 1.1.2.7.2

Combine the opposite terms in $1 - 3 a + 3 a - 1$ .

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Step 1.1.2.7.2.1

Add $- 3 a$ and $3 a$ .

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

Step 1.1.2.7.2.2

Add $1$ and $0$ .

$\frac{1 - 1}{lim_{x \to 1} {(x - 1)}^{2}}$

Step 1.1.2.7.2.3

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 1} {(x - 1)}^{2}}$

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

Move the exponent $2$ from ${(x - 1)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{x \to 1} x - 1)}^{2}}$

Step 1.1.3.1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{0}{{(lim_{x \to 1} x - lim_{x \to 1} 1)}^{2}}$

Step 1.1.3.1.3

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

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

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{0}{{(1 - 1 \cdot 1)}^{2}}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $- 1$ by $1$ .

$\frac{0}{{(1 - 1)}^{2}}$

Step 1.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0^{2}}$

Step 1.1.3.3.3

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

$\frac{0}{0}$

Step 1.1.3.3.4

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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Step 1.3.4.1

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

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

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 $n = 1$ .

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

Step 1.3.4.3

Multiply $- 3$ by $1$ .

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

Step 1.3.5

Since $3 a$ is constant with respect to $x$ , the derivative of $3 a$ with respect to $x$ is $0$ .

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

Step 1.3.6

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

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

Step 1.3.7

Simplify.

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Step 1.3.7.1

Combine terms.

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Step 1.3.7.1.1

Add $3 a x^{3 a - 1} - 3 a$ and $0$ .

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

Step 1.3.7.1.2

Add $3 a x^{3 a - 1} - 3 a$ and $0$ .

$lim_{x \to 1} \frac{3 a x^{3 a - 1} - 3 a}{\frac{d}{d x} [{(x - 1)}^{2}]}$

Step 1.3.7.2

Reorder terms.

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

Step 1.3.7.3

Reorder factors in $- 3 a + 3 x^{3 a - 1} a$ .

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

Step 1.3.8

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$lim_{x \to 1} \frac{- 3 a + 3 a x^{3 a - 1}}{\frac{d}{d x} [(x - 1) (x - 1)]}$

Step 1.3.9

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 1.3.9.1

Apply the distributive property.

$lim_{x \to 1} \frac{- 3 a + 3 a x^{3 a - 1}}{\frac{d}{d x} [x (x - 1) - 1 (x - 1)]}$

Step 1.3.9.2

Apply the distributive property.

$lim_{x \to 1} \frac{- 3 a + 3 a x^{3 a - 1}}{\frac{d}{d x} [x \cdot x + x \cdot - 1 - 1 (x - 1)]}$

Step 1.3.9.3

Apply the distributive property.

$lim_{x \to 1} \frac{- 3 a + 3 a x^{3 a - 1}}{\frac{d}{d x} [x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1]}$

Step 1.3.10

Simplify and combine like terms.

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Step 1.3.10.1

Simplify each term.

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Step 1.3.10.1.1

Multiply $x$ by $x$ .

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

Step 1.3.10.1.2

Move $- 1$ to the left of $x$ .

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

Step 1.3.10.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 1.3.10.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 1.3.10.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.3.10.2

Subtract $x$ from $- x$ .

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

Step 1.3.11

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

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

Step 1.3.12

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

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

Step 1.3.13

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

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

Step 1.3.14

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

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

Step 1.3.15

Multiply $- 2$ by $1$ .

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

Step 1.3.16

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

$lim_{x \to 1} \frac{- 3 a + 3 a x^{3 a - 1}}{2 x - 2 + 0}$

Step 1.3.17

Add $2 x - 2$ and $0$ .

$lim_{x \to 1} \frac{- 3 a + 3 a x^{3 a - 1}}{2 x - 2}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$\frac{lim_{x \to 1} - 3 a + 3 a x^{3 a - 1}}{lim_{x \to 1} 2 x - 2}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{- lim_{x \to 1} 3 a + lim_{x \to 1} 3 a x^{3 a - 1}}{lim_{x \to 1} 2 x - 2}$

Step 2.1.2.1.2

Evaluate the limit of $3 a$ which is constant as $x$ approaches $1$ .

$\frac{- 3 a + lim_{x \to 1} 3 a x^{3 a - 1}}{lim_{x \to 1} 2 x - 2}$

Step 2.1.2.1.3

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

$\frac{- 3 a + 3 a lim_{x \to 1} x^{3 a - 1}}{lim_{x \to 1} 2 x - 2}$

Step 2.1.2.1.4

Move the exponent $3 a - 1$ from $x^{3 a - 1}$ outside the limit using the Limits Power Rule.

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

Step 2.1.2.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{- 3 a + 3 a \cdot 1^{3 a - 1}}{lim_{x \to 1} 2 x - 2}$

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

One to any power is one.

$\frac{- 3 a + 3 a \cdot 1}{lim_{x \to 1} 2 x - 2}$

Step 2.1.2.3.1.2

Multiply $3$ by $1$ .

$\frac{- 3 a + 3 a}{lim_{x \to 1} 2 x - 2}$

Step 2.1.2.3.2

Add $- 3 a$ and $3 a$ .

$\frac{0}{lim_{x \to 1} 2 x - 2}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Evaluate the limit.

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Step 2.1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{0}{lim_{x \to 1} 2 x - lim_{x \to 1} 2}$

Step 2.1.3.1.2

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

$\frac{0}{2 lim_{x \to 1} x - lim_{x \to 1} 2}$

Step 2.1.3.1.3

Evaluate the limit of $2$ which is constant as $x$ approaches $1$ .

$\frac{0}{2 lim_{x \to 1} x - 1 \cdot 2}$

Step 2.1.3.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{0}{2 \cdot 1 - 1 \cdot 2}$

Step 2.1.3.3

Simplify the answer.

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Step 2.1.3.3.1

Simplify each term.

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Step 2.1.3.3.1.1

Multiply $2$ by $1$ .

$\frac{0}{2 - 1 \cdot 2}$

Step 2.1.3.3.1.2

Multiply $- 1$ by $2$ .

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

Step 2.1.3.3.2

Subtract $2$ from $2$ .

$\frac{0}{0}$

Step 2.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 2.1.3.4

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

Undefined

$\frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{0}{0}$

Step 2.2

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.

$lim_{x \to 1} \frac{- 3 a + 3 a x^{3 a - 1}}{2 x - 2} = lim_{x \to 1} \frac{\frac{d}{d x} [- 3 a + 3 a x^{3 a - 1}]}{\frac{d}{d x} [2 x - 2]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

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

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

Step 2.3.3

Since $- 3 a$ is constant with respect to $x$ , the derivative of $- 3 a$ with respect to $x$ is $0$ .

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

Step 2.3.4

Evaluate $\frac{d}{d x} [3 a x^{3 a - 1}]$ .

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Step 2.3.4.1

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

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

Step 2.3.4.2

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

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

Step 2.3.4.3

Subtract $1$ from $- 1$ .

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

Add $0$ and $3 a ((3 a - 1) x^{3 a - 2})$ .

$lim_{x \to 1} \frac{3 a ((3 a - 1) x^{3 a - 2})}{\frac{d}{d x} [2 x - 2]}$

Step 2.3.5.2

Reorder the factors of $3 a ((3 a - 1) x^{3 a - 2})$ .

$lim_{x \to 1} \frac{3 x^{3 a - 2} a (3 a - 1)}{\frac{d}{d x} [2 x - 2]}$

Step 2.3.6

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

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

Step 2.3.7

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

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Step 2.3.7.1

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

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

Step 2.3.7.2

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

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

Step 2.3.7.3

Multiply $2$ by $1$ .

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

Step 2.3.8

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 2.3.9

Add $2$ and $0$ .

$lim_{x \to 1} \frac{3 x^{3 a - 2} a (3 a - 1)}{2}$

Step 3

Evaluate the limit.

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Step 3.1

Move the term $\frac{3 a (3 a - 1)}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 a (3 a - 1)}{2} lim_{x \to 1} x^{3 a - 2}$

Step 3.2

Move the exponent $3 a - 2$ from $x^{3 a - 2}$ outside the limit using the Limits Power Rule.

$\frac{3 a (3 a - 1)}{2} {(lim_{x \to 1} x)}^{3 a - 2}$

Step 4

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{3 a (3 a - 1)}{2} \cdot 1^{3 a - 2}$

Step 5

Simplify the answer.

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Step 5.1

One to any power is one.

$\frac{3 a (3 a - 1)}{2} \cdot 1$

Step 5.2

Multiply $\frac{3 a (3 a - 1)}{2}$ by $1$ .

$\frac{3 a (3 a - 1)}{2}$