Calculus Examples

$lim_{x \to 0} \frac{17 x}{\ln ({(x^{2} - 2 x + 1)}^{3})}$

Step 1

Move the term $17$ outside of the limit because it is constant with respect to $x$ .

$17 lim_{x \to 0} \frac{x}{\ln ({(x^{2} - 2 x + 1)}^{3})}$

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.

$17 \frac{lim_{x \to 0} x}{lim_{x \to 0} \ln ({(x^{2} - 2 x + 1)}^{3})}$

Step 2.1.2

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

$17 \frac{0}{lim_{x \to 0} \ln ({(x^{2} - 2 x + 1)}^{3})}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Move the limit inside the logarithm.

$17 \frac{0}{\ln (lim_{x \to 0} {(x^{2} - 2 x + 1)}^{3})}$

Step 2.1.3.2

Move the exponent $3$ from ${(x^{2} - 2 x + 1)}^{3}$ outside the limit using the Limits Power Rule.

$17 \frac{0}{\ln ({(lim_{x \to 0} x^{2} - 2 x + 1)}^{3})}$

Step 2.1.3.3

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

$17 \frac{0}{\ln ({(lim_{x \to 0} x^{2} - lim_{x \to 0} 2 x + lim_{x \to 0} 1)}^{3})}$

Step 2.1.3.4

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

$17 \frac{0}{\ln ({({(lim_{x \to 0} x)}^{2} - lim_{x \to 0} 2 x + lim_{x \to 0} 1)}^{3})}$

Step 2.1.3.5

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

$17 \frac{0}{\ln ({({(lim_{x \to 0} x)}^{2} - 2 lim_{x \to 0} x + lim_{x \to 0} 1)}^{3})}$

Step 2.1.3.6

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

$17 \frac{0}{\ln ({({(lim_{x \to 0} x)}^{2} - 2 lim_{x \to 0} x + 1)}^{3})}$

Step 2.1.3.7

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

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Step 2.1.3.7.1

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

$17 \frac{0}{\ln ({(0^{2} - 2 lim_{x \to 0} x + 1)}^{3})}$

Step 2.1.3.7.2

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

$17 \frac{0}{\ln ({(0^{2} - 2 \cdot 0 + 1)}^{3})}$

Step 2.1.3.8

Simplify the answer.

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Step 2.1.3.8.1

Simplify each term.

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Step 2.1.3.8.1.1

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

$17 \frac{0}{\ln ({(0 - 2 \cdot 0 + 1)}^{3})}$

Step 2.1.3.8.1.2

Multiply $- 2$ by $0$ .

$17 \frac{0}{\ln ({(0 + 0 + 1)}^{3})}$

Step 2.1.3.8.2

Add $0$ and $0$ .

$17 \frac{0}{\ln ({(0 + 1)}^{3})}$

Step 2.1.3.8.3

Add $0$ and $1$ .

$17 \frac{0}{\ln (1^{3})}$

Step 2.1.3.8.4

One to any power is one.

$17 \frac{0}{\ln (1)}$

Step 2.1.3.8.5

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

$17 (\frac{0}{0})$

Step 2.1.3.8.6

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

Undefined

$17 (\frac{0}{0})$

Step 2.1.3.9

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

Undefined

$17 (\frac{0}{0})$

Step 2.1.4

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

Undefined

$17 (\frac{0}{0})$

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$17 lim_{x \to 0} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [\ln ({(x^{2} - 2 x + 1)}^{3})]}$

Step 2.3.2

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

$17 lim_{x \to 0} \frac{1}{\frac{d}{d x} [\ln ({(x^{2} - 2 x + 1)}^{3})]}$

Step 2.3.3

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 $g (x) = {(x^{2} - 2 x + 1)}^{3}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u_{1}$ as ${(x^{2} - 2 x + 1)}^{3}$ .

$17 lim_{x \to 0} \frac{1}{\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [{(x^{2} - 2 x + 1)}^{3}]}$

Step 2.3.3.2

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

$17 lim_{x \to 0} \frac{1}{\frac{1}{u_{1}} \frac{d}{d x} [{(x^{2} - 2 x + 1)}^{3}]}$

Step 2.3.3.3

Replace all occurrences of $u_{1}$ with ${(x^{2} - 2 x + 1)}^{3}$ .

$17 lim_{x \to 0} \frac{1}{\frac{1}{{(x^{2} - 2 x + 1)}^{3}} \frac{d}{d x} [{(x^{2} - 2 x + 1)}^{3}]}$

Step 2.3.4

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 $g (x) = x^{2} - 2 x + 1$ .

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Step 2.3.4.1

To apply the Chain Rule, set $u_{2}$ as $x^{2} - 2 x + 1$ .

$17 lim_{x \to 0} \frac{1}{\frac{1}{{(x^{2} - 2 x + 1)}^{3}} \frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d x} [x^{2} - 2 x + 1]}$

Step 2.3.4.2

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

$17 lim_{x \to 0} \frac{1}{\frac{1}{{(x^{2} - 2 x + 1)}^{3}} \cdot 3 {u_{2}}^{2} \frac{d}{d x} [x^{2} - 2 x + 1]}$

Step 2.3.4.3

Replace all occurrences of $u_{2}$ with $x^{2} - 2 x + 1$ .

$17 lim_{x \to 0} \frac{1}{\frac{1}{{(x^{2} - 2 x + 1)}^{3}} \cdot 3 {(x^{2} - 2 x + 1)}^{2} \frac{d}{d x} [x^{2} - 2 x + 1]}$

Step 2.3.5

Combine $3$ and $\frac{1}{{(x^{2} - 2 x + 1)}^{3}}$ .

$17 lim_{x \to 0} \frac{1}{\frac{3}{{(x^{2} - 2 x + 1)}^{3}} {(x^{2} - 2 x + 1)}^{2} \frac{d}{d x} [x^{2} - 2 x + 1]}$

Step 2.3.6

Combine ${(x^{2} - 2 x + 1)}^{2}$ and $\frac{3}{{(x^{2} - 2 x + 1)}^{3}}$ .

$17 lim_{x \to 0} \frac{1}{\frac{{(x^{2} - 2 x + 1)}^{2} \cdot 3}{{(x^{2} - 2 x + 1)}^{3}} \frac{d}{d x} [x^{2} - 2 x + 1]}$

Step 2.3.7

Move $3$ to the left of ${(x^{2} - 2 x + 1)}^{2}$ .

$17 lim_{x \to 0} \frac{1}{\frac{3 \cdot {(x^{2} - 2 x + 1)}^{2}}{{(x^{2} - 2 x + 1)}^{3}} \frac{d}{d x} [x^{2} - 2 x + 1]}$

Step 2.3.8

Cancel the common factor of ${(x^{2} - 2 x + 1)}^{2}$ and ${(x^{2} - 2 x + 1)}^{3}$ .

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Step 2.3.8.1

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

$17 lim_{x \to 0} \frac{1}{\frac{{(x^{2} - 2 x + 1)}^{2} \cdot 3}{{(x^{2} - 2 x + 1)}^{3}} \frac{d}{d x} [x^{2} - 2 x + 1]}$

Step 2.3.8.2

Cancel the common factors.

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Step 2.3.8.2.1

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

$17 lim_{x \to 0} \frac{1}{\frac{{(x^{2} - 2 x + 1)}^{2} \cdot 3}{{(x^{2} - 2 x + 1)}^{2} (x^{2} - 2 x + 1)} \frac{d}{d x} [x^{2} - 2 x + 1]}$

Step 2.3.8.2.2

Cancel the common factor.

$17 lim_{x \to 0} \frac{1}{\frac{{(x^{2} - 2 x + 1)}^{2} \cdot 3}{{(x^{2} - 2 x + 1)}^{2} (x^{2} - 2 x + 1)} \frac{d}{d x} [x^{2} - 2 x + 1]}$

Step 2.3.8.2.3

Rewrite the expression.

$17 lim_{x \to 0} \frac{1}{\frac{3}{x^{2} - 2 x + 1} \frac{d}{d x} [x^{2} - 2 x + 1]}$

Step 2.3.9

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

$17 lim_{x \to 0} \frac{1}{\frac{3}{x^{2} - 2 x + 1} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [1])}$

Step 2.3.10

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

$17 lim_{x \to 0} \frac{1}{\frac{3}{x^{2} - 2 x + 1} (2 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [1])}$

Step 2.3.11

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

$17 lim_{x \to 0} \frac{1}{\frac{3}{x^{2} - 2 x + 1} (2 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [1])}$

Step 2.3.12

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

$17 lim_{x \to 0} \frac{1}{\frac{3}{x^{2} - 2 x + 1} (2 x - 2 \cdot 1 + \frac{d}{d x} [1])}$

Step 2.3.13

Multiply $- 2$ by $1$ .

$17 lim_{x \to 0} \frac{1}{\frac{3}{x^{2} - 2 x + 1} (2 x - 2 + \frac{d}{d x} [1])}$

Step 2.3.14

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

$17 lim_{x \to 0} \frac{1}{\frac{3}{x^{2} - 2 x + 1} (2 x - 2 + 0)}$

Step 2.3.15

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

$17 lim_{x \to 0} \frac{1}{\frac{3}{x^{2} - 2 x + 1} (2 x - 2)}$

Step 2.3.16

Simplify.

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Step 2.3.16.1

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

$17 lim_{x \to 0} \frac{1}{(2 x - 2) \frac{3}{x^{2} - 2 x + 1}}$

Step 2.3.16.2

Factor using the perfect square rule.

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Step 2.3.16.2.1

Rewrite $1$ as $1^{2}$ .

$17 lim_{x \to 0} \frac{1}{(2 x - 2) \frac{3}{x^{2} - 2 x + 1^{2}}}$

Step 2.3.16.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 2.3.16.2.3

Rewrite the polynomial.

$17 lim_{x \to 0} \frac{1}{(2 x - 2) \frac{3}{x^{2} - 2 \cdot x \cdot 1 + 1^{2}}}$

Step 2.3.16.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

$17 lim_{x \to 0} \frac{1}{(2 x - 2) \frac{3}{{(x - 1)}^{2}}}$

Step 2.3.16.3

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

$17 lim_{x \to 0} \frac{1}{\frac{(2 x - 2) \cdot 3}{{(x - 1)}^{2}}}$

Step 2.3.16.4

Simplify the numerator.

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Step 2.3.16.4.1

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

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Step 2.3.16.4.1.1

Factor $2$ out of $2 x$ .

$17 lim_{x \to 0} \frac{1}{\frac{(2 (x) - 2) \cdot 3}{{(x - 1)}^{2}}}$

Step 2.3.16.4.1.2

Factor $2$ out of $- 2$ .

$17 lim_{x \to 0} \frac{1}{\frac{(2 (x) + 2 (- 1)) \cdot 3}{{(x - 1)}^{2}}}$

Step 2.3.16.4.1.3

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

$17 lim_{x \to 0} \frac{1}{\frac{2 (x - 1) \cdot 3}{{(x - 1)}^{2}}}$

Step 2.3.16.4.2

Multiply $3$ by $2$ .

$17 lim_{x \to 0} \frac{1}{\frac{6 (x - 1)}{{(x - 1)}^{2}}}$

Step 2.3.16.5

Cancel the common factor of $x - 1$ and ${(x - 1)}^{2}$ .

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Step 2.3.16.5.1

Factor $x - 1$ out of $6 (x - 1)$ .

$17 lim_{x \to 0} \frac{1}{\frac{(x - 1) \cdot 6}{{(x - 1)}^{2}}}$

Step 2.3.16.5.2

Cancel the common factors.

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Step 2.3.16.5.2.1

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

$17 lim_{x \to 0} \frac{1}{\frac{(x - 1) \cdot 6}{(x - 1) (x - 1)}}$

Step 2.3.16.5.2.2

Cancel the common factor.

$17 lim_{x \to 0} \frac{1}{\frac{(x - 1) \cdot 6}{(x - 1) (x - 1)}}$

Step 2.3.16.5.2.3

Rewrite the expression.

$17 lim_{x \to 0} \frac{1}{\frac{6}{x - 1}}$

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$17 lim_{x \to 0} 1 \frac{x - 1}{6}$

Step 2.5

Multiply $\frac{x - 1}{6}$ by $1$ .

$17 lim_{x \to 0} \frac{x - 1}{6}$

Step 3

Evaluate the limit.

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Step 3.1

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

$17 (\frac{1}{6}) lim_{x \to 0} x - 1$

Step 3.2

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

$17 (\frac{1}{6}) (lim_{x \to 0} x - lim_{x \to 0} 1)$

Step 3.3

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

$17 (\frac{1}{6}) (lim_{x \to 0} x - 1 \cdot 1)$

Step 4

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

$17 (\frac{1}{6}) (0 - 1 \cdot 1)$

Step 5

Simplify the answer.

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Step 5.1

Combine $17$ and $\frac{1}{6}$ .

$\frac{17}{6} (0 - 1 \cdot 1)$

Step 5.2

Multiply $- 1$ by $1$ .

$\frac{17}{6} (0 - 1)$

Step 5.3

Subtract $1$ from $0$ .

$\frac{17}{6} \cdot - 1$

Step 5.4

Multiply $\frac{17}{6} \cdot - 1$ .

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Step 5.4.1

Combine $\frac{17}{6}$ and $- 1$ .

$\frac{17 \cdot - 1}{6}$

Step 5.4.2

Multiply $17$ by $- 1$ .

$\frac{- 17}{6}$

Step 5.5

Move the negative in front of the fraction.

$- \frac{17}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{17}{6}$

Decimal Form:

$- 2.8\overline{3}$