Calculus Examples

$lim_{x \to 1} \frac{\ln (x)}{20 x - x^{2} - 19}$

Step 1

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

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Step 1.1

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

$\frac{lim_{x \to 1} \ln (x)}{lim_{x \to 1} 20 x - x^{2} - 19}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Move the limit inside the logarithm.

$\frac{\ln (lim_{x \to 1} x)}{lim_{x \to 1} 20 x - x^{2} - 19}$

Step 1.2.2

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

$\frac{\ln (1)}{lim_{x \to 1} 20 x - x^{2} - 19}$

Step 1.2.3

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

$\frac{0}{lim_{x \to 1} 20 x - x^{2} - 19}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

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

$\frac{0}{lim_{x \to 1} 20 x - lim_{x \to 1} x^{2} - lim_{x \to 1} 19}$

Step 1.3.2

Move the term $20$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{20 lim_{x \to 1} x - lim_{x \to 1} x^{2} - lim_{x \to 1} 19}$

Step 1.3.3

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

$\frac{0}{20 lim_{x \to 1} x - {(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 19}$

Step 1.3.4

Evaluate the limit of $19$ which is constant as $x$ approaches $1$ .

$\frac{0}{20 lim_{x \to 1} x - {(lim_{x \to 1} x)}^{2} - 1 \cdot 19}$

Step 1.3.5

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

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Step 1.3.5.1

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

$\frac{0}{20 \cdot 1 - {(lim_{x \to 1} x)}^{2} - 1 \cdot 19}$

Step 1.3.5.2

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

$\frac{0}{20 \cdot 1 - 1^{2} - 1 \cdot 19}$

Step 1.3.6

Simplify the answer.

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Step 1.3.6.1

Simplify each term.

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Step 1.3.6.1.1

Multiply $20$ by $1$ .

$\frac{0}{20 - 1^{2} - 1 \cdot 19}$

Step 1.3.6.1.2

One to any power is one.

$\frac{0}{20 - 1 \cdot 1 - 1 \cdot 19}$

Step 1.3.6.1.3

Multiply $- 1$ by $1$ .

$\frac{0}{20 - 1 - 1 \cdot 19}$

Step 1.3.6.1.4

Multiply $- 1$ by $19$ .

$\frac{0}{20 - 1 - 19}$

Step 1.3.6.2

Subtract $1$ from $20$ .

$\frac{0}{19 - 19}$

Step 1.3.6.3

Subtract $19$ from $19$ .

$\frac{0}{0}$

Step 1.3.6.4

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

Undefined

$\frac{0}{0}$

Step 1.3.7

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

$lim_{x \to 1} \frac{\ln (x)}{20 x - x^{2} - 19} = lim_{x \to 1} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [20 x - x^{2} - 19]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 1} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [20 x - x^{2} - 19]}$

Step 3.2

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

$lim_{x \to 1} \frac{\frac{1}{x}}{\frac{d}{d x} [20 x - x^{2} - 19]}$

Step 3.3

By the Sum Rule, the derivative of $20 x - x^{2} - 19$ with respect to $x$ is $\frac{d}{d x} [20 x] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 19]$ .

$lim_{x \to 1} \frac{\frac{1}{x}}{\frac{d}{d x} [20 x] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 19]}$

Step 3.4

Evaluate $\frac{d}{d x} [20 x]$ .

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Step 3.4.1

Since $20$ is constant with respect to $x$ , the derivative of $20 x$ with respect to $x$ is $20 \frac{d}{d x} [x]$ .

$lim_{x \to 1} \frac{\frac{1}{x}}{20 \frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 19]}$

Step 3.4.2

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

$lim_{x \to 1} \frac{\frac{1}{x}}{20 \cdot 1 + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 19]}$

Step 3.4.3

Multiply $20$ by $1$ .

$lim_{x \to 1} \frac{\frac{1}{x}}{20 + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 19]}$

Step 3.5

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

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Step 3.5.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$lim_{x \to 1} \frac{\frac{1}{x}}{20 - \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 19]}$

Step 3.5.2

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

$lim_{x \to 1} \frac{\frac{1}{x}}{20 - (2 x) + \frac{d}{d x} [- 19]}$

Step 3.5.3

Multiply $2$ by $- 1$ .

$lim_{x \to 1} \frac{\frac{1}{x}}{20 - 2 x + \frac{d}{d x} [- 19]}$

Step 3.6

Since $- 19$ is constant with respect to $x$ , the derivative of $- 19$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{\frac{1}{x}}{20 - 2 x + 0}$

Step 3.7

Simplify.

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Step 3.7.1

Add $20 - 2 x$ and $0$ .

$lim_{x \to 1} \frac{\frac{1}{x}}{20 - 2 x}$

Step 3.7.2

Reorder terms.

$lim_{x \to 1} \frac{\frac{1}{x}}{- 2 x + 20}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 1} \frac{1}{x} \cdot \frac{1}{- 2 x + 20}$

Step 5

Multiply $\frac{1}{x}$ by $\frac{1}{- 2 x + 20}$ .

$lim_{x \to 1} \frac{1}{x (- 2 x + 20)}$

Step 6

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

$\frac{lim_{x \to 1} 1}{lim_{x \to 1} x (- 2 x + 20)}$

Step 7

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

$\frac{1}{lim_{x \to 1} x (- 2 x + 20)}$

Step 8

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

$\frac{1}{lim_{x \to 1} x \cdot lim_{x \to 1} - 2 x + 20}$

Step 9

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

$\frac{1}{lim_{x \to 1} x \cdot (- lim_{x \to 1} 2 x + lim_{x \to 1} 20)}$

Step 10

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

$\frac{1}{lim_{x \to 1} x \cdot (- 2 lim_{x \to 1} x + lim_{x \to 1} 20)}$

Step 11

Evaluate the limit of $20$ which is constant as $x$ approaches $1$ .

$\frac{1}{lim_{x \to 1} x \cdot (- 2 lim_{x \to 1} x + 20)}$

Step 12

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

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Step 12.1

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

$\frac{1}{1 \cdot (- 2 lim_{x \to 1} x + 20)}$

Step 12.2

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

$\frac{1}{1 \cdot (- 2 \cdot 1 + 20)}$

Step 13

Simplify the answer.

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Step 13.1

Cancel the common factor of $1$ .

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Step 13.1.1

Cancel the common factor.

$\frac{1}{1 \cdot (- 2 \cdot 1 + 20)}$

Step 13.1.2

Rewrite the expression.

$\frac{1}{- 2 \cdot 1 + 20}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Multiply $- 2$ by $1$ .

$\frac{1}{- 2 + 20}$

Step 13.2.2

Add $- 2$ and $20$ .

$\frac{1}{18}$