Calculus Examples

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

Step 1

Combine terms.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Write each expression with a common denominator of $(x - 1) (x^{2} - 3 x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{1}{x - 1}$ by $\frac{x^{2} - 3 x + 2}{x^{2} - 3 x + 2}$ .

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

Step 1.3.2

Multiply $\frac{1}{x^{2} - 3 x + 2}$ by $\frac{x - 1}{x - 1}$ .

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

Step 1.3.3

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

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Add $- 3 x$ and $x$ .

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

Step 1.6

Subtract $1$ from $2$ .

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

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.

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

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

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

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

Step 2.1.2.5

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

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Step 2.1.2.5.1

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

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

Step 2.1.2.5.2

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

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

Step 2.1.2.6

Simplify the answer.

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Step 2.1.2.6.1

Simplify each term.

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Step 2.1.2.6.1.1

One to any power is one.

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

Step 2.1.2.6.1.2

Multiply $- 2$ by $1$ .

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

Step 2.1.2.6.2

Subtract $2$ from $1$ .

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

Step 2.1.2.6.3

Add $- 1$ and $1$ .

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

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

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

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

Step 2.1.3.5

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

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

Step 2.1.3.6

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

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

Step 2.1.3.7

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

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

Step 2.1.3.8

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

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Step 2.1.3.8.1

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

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

Step 2.1.3.8.2

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

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

Step 2.1.3.8.3

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

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

Step 2.1.3.9

Simplify the answer.

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Step 2.1.3.9.1

Multiply $- 1$ by $1$ .

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

Step 2.1.3.9.2

Subtract $1$ from $1$ .

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

Step 2.1.3.9.3

Simplify each term.

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Step 2.1.3.9.3.1

One to any power is one.

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

Step 2.1.3.9.3.2

Multiply $- 3$ by $1$ .

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

Step 2.1.3.9.4

Subtract $3$ from $1$ .

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

Step 2.1.3.9.5

Add $- 2$ and $2$ .

$\frac{0}{0 \cdot 0}$

Step 2.1.3.9.6

Multiply $0$ by $0$ .

$\frac{0}{0}$

Step 2.1.3.9.7

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

Undefined

$\frac{0}{0}$

Step 2.1.3.10

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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Step 2.3.4.1

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

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

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 $n = 1$ .

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

Step 2.3.4.3

Multiply $- 2$ by $1$ .

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

Step 2.3.5

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

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

Step 2.3.6

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

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

Step 2.3.7

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x - 1$ and $g (x) = x^{2} - 3 x + 2$ .

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

Step 2.3.8

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

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

Step 2.3.9

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

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

Step 2.3.10

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

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

Step 2.3.11

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

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

Step 2.3.12

Multiply $- 3$ by $1$ .

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

Step 2.3.13

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

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

Step 2.3.14

Add $2 x - 3$ and $0$ .

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

Step 2.3.15

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

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

Step 2.3.16

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

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

Step 2.3.17

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

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

Step 2.3.18

Add $1$ and $0$ .

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

Step 2.3.19

Multiply $x^{2} - 3 x + 2$ by $1$ .

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

Step 2.3.20

Simplify.

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Step 2.3.20.1

Apply the distributive property.

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

Step 2.3.20.2

Apply the distributive property.

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

Step 2.3.20.3

Apply the distributive property.

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

Step 2.3.20.4

Combine terms.

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Step 2.3.20.4.1

Raise $x$ to the power of $1$ .

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

Step 2.3.20.4.2

Raise $x$ to the power of $1$ .

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

Step 2.3.20.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3.20.4.4

Add $1$ and $1$ .

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

Step 2.3.20.4.5

Multiply $2$ by $- 1$ .

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

Step 2.3.20.4.6

Move $- 3$ to the left of $x$ .

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

Step 2.3.20.4.7

Multiply $- 1$ by $- 3$ .

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

Step 2.3.20.4.8

Subtract $3 x$ from $- 2 x$ .

$lim_{x \to 1} \frac{2 x - 2}{2 x^{2} - 5 x + 3 + x^{2} - 3 x + 2}$

Step 2.3.20.4.9

Add $2 x^{2}$ and $x^{2}$ .

$lim_{x \to 1} \frac{2 x - 2}{3 x^{2} - 5 x + 3 - 3 x + 2}$

Step 2.3.20.4.10

Subtract $3 x$ from $- 5 x$ .

$lim_{x \to 1} \frac{2 x - 2}{3 x^{2} - 8 x + 3 + 2}$

Step 2.3.20.4.11

Add $3$ and $2$ .

$lim_{x \to 1} \frac{2 x - 2}{3 x^{2} - 8 x + 5}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

$\frac{lim_{x \to 1} 2 x - 2}{lim_{x \to 1} 3 x^{2} - 8 x + 5}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Evaluate the limit.

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Step 3.1.2.1.1

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

$\frac{lim_{x \to 1} 2 x - lim_{x \to 1} 2}{lim_{x \to 1} 3 x^{2} - 8 x + 5}$

Step 3.1.2.1.2

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

$\frac{2 lim_{x \to 1} x - lim_{x \to 1} 2}{lim_{x \to 1} 3 x^{2} - 8 x + 5}$

Step 3.1.2.1.3

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

$\frac{2 lim_{x \to 1} x - 1 \cdot 2}{lim_{x \to 1} 3 x^{2} - 8 x + 5}$

Step 3.1.2.2

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

$\frac{2 \cdot 1 - 1 \cdot 2}{lim_{x \to 1} 3 x^{2} - 8 x + 5}$

Step 3.1.2.3

Simplify the answer.

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Step 3.1.2.3.1

Simplify each term.

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Step 3.1.2.3.1.1

Multiply $2$ by $1$ .

$\frac{2 - 1 \cdot 2}{lim_{x \to 1} 3 x^{2} - 8 x + 5}$

Step 3.1.2.3.1.2

Multiply $- 1$ by $2$ .

$\frac{2 - 2}{lim_{x \to 1} 3 x^{2} - 8 x + 5}$

Step 3.1.2.3.2

Subtract $2$ from $2$ .

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

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

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

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.3.4

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

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

Step 3.1.3.5

Evaluate the limit of $5$ which is constant as $x$ approaches $1$ .

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

Step 3.1.3.6

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

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Step 3.1.3.6.1

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

$\frac{0}{3 \cdot 1^{2} - 8 lim_{x \to 1} x + 5}$

Step 3.1.3.6.2

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

$\frac{0}{3 \cdot 1^{2} - 8 \cdot 1 + 5}$

Step 3.1.3.7

Simplify the answer.

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Step 3.1.3.7.1

Simplify each term.

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Step 3.1.3.7.1.1

One to any power is one.

$\frac{0}{3 \cdot 1 - 8 \cdot 1 + 5}$

Step 3.1.3.7.1.2

Multiply $3$ by $1$ .

$\frac{0}{3 - 8 \cdot 1 + 5}$

Step 3.1.3.7.1.3

Multiply $- 8$ by $1$ .

$\frac{0}{3 - 8 + 5}$

Step 3.1.3.7.2

Subtract $8$ from $3$ .

$\frac{0}{- 5 + 5}$

Step 3.1.3.7.3

Add $- 5$ and $5$ .

$\frac{0}{0}$

Step 3.1.3.7.4

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

Undefined

$\frac{0}{0}$

Step 3.1.3.8

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

Undefined

$\frac{0}{0}$

Step 3.1.4

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

Undefined

$\frac{0}{0}$

Step 3.2

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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

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

Step 3.3.3

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

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Step 3.3.3.1

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

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

Step 3.3.3.2

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

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

Step 3.3.3.3

Multiply $2$ by $1$ .

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

Step 3.3.4

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

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

Step 3.3.5

Add $2$ and $0$ .

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

Step 3.3.6

By the Sum Rule, the derivative of $3 x^{2} - 8 x + 5$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [5]$ .

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

Step 3.3.7

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

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Step 3.3.7.1

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

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

Step 3.3.7.2

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

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

Step 3.3.7.3

Multiply $2$ by $3$ .

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

Step 3.3.8

Evaluate $\frac{d}{d x} [- 8 x]$ .

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Step 3.3.8.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

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

Step 3.3.8.2

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

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

Step 3.3.8.3

Multiply $- 8$ by $1$ .

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

Step 3.3.9

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

$lim_{x \to 1} \frac{2}{6 x - 8 + 0}$

Step 3.3.10

Add $6 x - 8$ and $0$ .

$lim_{x \to 1} \frac{2}{6 x - 8}$

Step 3.4

Cancel the common factor of $2$ and $6 x - 8$ .

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Step 3.4.1

Factor $2$ out of $2$ .

$lim_{x \to 1} \frac{2 (1)}{6 x - 8}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $6 x$ .

$lim_{x \to 1} \frac{2 (1)}{2 (3 x) - 8}$

Step 3.4.2.2

Factor $2$ out of $- 8$ .

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

Step 3.4.2.3

Factor $2$ out of $2 (3 x) + 2 (- 4)$ .

$lim_{x \to 1} \frac{2 (1)}{2 (3 x - 4)}$

Step 3.4.2.4

Cancel the common factor.

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

Step 3.4.2.5

Rewrite the expression.

$lim_{x \to 1} \frac{1}{3 x - 4}$

Step 4

Evaluate the limit.

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Step 4.1

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

$\frac{lim_{x \to 1} 1}{lim_{x \to 1} 3 x - 4}$

Step 4.2

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

$\frac{1}{lim_{x \to 1} 3 x - 4}$

Step 4.3

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

$\frac{1}{lim_{x \to 1} 3 x - lim_{x \to 1} 4}$

Step 4.4

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

$\frac{1}{3 lim_{x \to 1} x - lim_{x \to 1} 4}$

Step 4.5

Evaluate the limit of $4$ which is constant as $x$ approaches $1$ .

$\frac{1}{3 lim_{x \to 1} x - 1 \cdot 4}$

Step 5

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

$\frac{1}{3 \cdot 1 - 1 \cdot 4}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the denominator.

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Step 6.1.1

Multiply $3$ by $1$ .

$\frac{1}{3 - 1 \cdot 4}$

Step 6.1.2

Multiply $- 1$ by $4$ .

$\frac{1}{3 - 4}$

Step 6.1.3

Subtract $4$ from $3$ .

$\frac{1}{- 1}$

Step 6.2

Divide $1$ by $- 1$ .

$- 1$