Calculus Examples

$\ln (1 + \frac{1}{x})$

Step 1

Find the asymptotes.

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Set the argument of the logarithm equal to zero.

$1 + \frac{1}{x} = 0$

Solve for $x$ .

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Subtract $1$ from both sides of the equation.

$\frac{1}{x} = - 1$

Find the LCD of the terms in the equation.

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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

The LCM of one and any expression is the expression.

$x$

Multiply each term in $\frac{1}{x} = - 1$ by $x$ to eliminate the fractions.

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Multiply each term in $\frac{1}{x} = - 1$ by $x$ .

$\frac{1}{x} x = - x$

Simplify the left side.

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Cancel the common factor of $x$ .

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Cancel the common factor.

$\frac{1}{x} x = - x$

Rewrite the expression.

$1 = - x$

Solve the equation.

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Rewrite the equation as $- x = 1$ .

$- x = 1$

Divide each term in $- x = 1$ by $- 1$ and simplify.

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Divide each term in $- x = 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{1}{- 1}$

Simplify the left side.

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Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{1}{- 1}$

Divide $x$ by $1$ .

$x = \frac{1}{- 1}$

Simplify the right side.

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Divide $1$ by $- 1$ .

$x = - 1$

The vertical asymptote occurs at $x = - 1$ .

Vertical Asymptote: $x = - 1$

Step 2

Find the point at $x = - 2$ .

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Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = \ln (1 + \frac{1}{- 2})$

Simplify the result.

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Move the negative in front of the fraction.

$f (- 2) = \ln (1 - \frac{1}{2})$

Write $1$ as a fraction with a common denominator.

$f (- 2) = \ln (\frac{2}{2} - \frac{1}{2})$

Combine the numerators over the common denominator.

$f (- 2) = \ln (\frac{2 - 1}{2})$

Subtract $1$ from $2$ .

$f (- 2) = \ln (\frac{1}{2})$

The final answer is $\ln (\frac{1}{2})$ .

$\ln (\frac{1}{2})$

Convert $\ln (\frac{1}{2})$ to decimal.

$= - 0.69314718$

Step 3

Find the point at $x = - 3$ .

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Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = \ln (1 + \frac{1}{- 3})$

Simplify the result.

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Move the negative in front of the fraction.

$f (- 3) = \ln (1 - \frac{1}{3})$

Write $1$ as a fraction with a common denominator.

$f (- 3) = \ln (\frac{3}{3} - \frac{1}{3})$

Combine the numerators over the common denominator.

$f (- 3) = \ln (\frac{3 - 1}{3})$

Subtract $1$ from $3$ .

$f (- 3) = \ln (\frac{2}{3})$

The final answer is $\ln (\frac{2}{3})$ .

$\ln (\frac{2}{3})$

Convert $\ln (\frac{2}{3})$ to decimal.

$= - 0.4054651$

Step 4

Find the point at $x = - 4$ .

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Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = \ln (1 + \frac{1}{- 4})$

Simplify the result.

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Move the negative in front of the fraction.

$f (- 4) = \ln (1 - \frac{1}{4})$

Write $1$ as a fraction with a common denominator.

$f (- 4) = \ln (\frac{4}{4} - \frac{1}{4})$

Combine the numerators over the common denominator.

$f (- 4) = \ln (\frac{4 - 1}{4})$

Subtract $1$ from $4$ .

$f (- 4) = \ln (\frac{3}{4})$

The final answer is $\ln (\frac{3}{4})$ .

$\ln (\frac{3}{4})$

Convert $\ln (\frac{3}{4})$ to decimal.

$= - 0.28768207$

Step 5

The log function can be graphed using the vertical asymptote at $x = - 1$ and the points $(- 2, - 0.69314718), (- 3, - 0.4054651), (- 4, - 0.28768207)$ .

Vertical Asymptote: $x = - 1$

$\begin{array}{c} x & y \\ - 4 & - 0.288 \\ - 3 & - 0.405 \\ - 2 & - 0.693 \end{array}$

Step 6