Trigonometry Examples

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

Step 1

Find the asymptotes.

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Step 1.1

Find where the expression $\frac{2 e - x - x \ln (x)}{x}$ is undefined.

$x \leq 0$

Step 1.2

Since $\frac{2 e - x - x \ln (x)}{x}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the left and $\frac{2 e - x - x \ln (x)}{x}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the right, then $x = 0$ is a vertical asymptote.

$x = 0$

Step 1.3

Ignoring the logarithm, consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 1.4

Find $n$ and $m$ .

$n = 1$

$m = 1$

Step 1.5

Since $n = m$ , the horizontal asymptote is the line $y = \frac{a}{b}$ where $a = - 2$ and $b = 1$ .

$y = - 2$

Step 1.6

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

Step 1.7

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = - 2$

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = - 2$

Step 2

Find the point at $x = 1$ .

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Step 2.1

Replace the variable $x$ with $1$ in the expression.

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

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $- 1$ by $1$ .

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

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Divide $2 e - 1$ by $1$ .

$f (1) = 2 e - 1 - \ln (1)$

Step 2.2.2.2

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

$f (1) = 2 e - 1 - 0$

Step 2.2.2.3

Multiply $- 1$ by $0$ .

$f (1) = 2 e - 1 + 0$

Step 2.2.3

Add $2 e$ and $0$ .

$f (1) = 2 e - 1$

Step 2.2.4

The final answer is $2 e - 1$ .

$2 e - 1$

Step 2.3

Convert $2 e - 1$ to decimal.

$y = 4.43656365$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $- 1$ by $2$ .

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

Step 3.2.2

Cancel the common factor of $2 e - 2$ and $2$ .

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Step 3.2.2.1

Factor $2$ out of $2 e$ .

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

Step 3.2.2.2

Factor $2$ out of $- 2$ .

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

Step 3.2.2.3

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

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

Step 3.2.2.4

Cancel the common factors.

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Step 3.2.2.4.1

Factor $2$ out of $2$ .

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

Step 3.2.2.4.2

Cancel the common factor.

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

Step 3.2.2.4.3

Rewrite the expression.

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

Step 3.2.2.4.4

Divide $e - 1$ by $1$ .

$f (2) = e - 1 - \ln (2)$

Step 3.2.3

The final answer is $e - 1 - \ln (2)$ .

$e - 1 - \ln (2)$

Step 3.3

Convert $e - 1 - \ln (2)$ to decimal.

$y = 1.02513464$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $- 1$ by $3$ .

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

Step 4.2.2

To write $- \ln (3)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4.2.3

Combine fractions.

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Step 4.2.3.1

Combine $- \ln (3)$ and $\frac{3}{3}$ .

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

Step 4.2.3.2

Combine the numerators over the common denominator.

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

Step 4.2.4

Simplify the numerator.

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Step 4.2.4.1

Multiply $- \ln (3) \cdot 3$ .

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Step 4.2.4.1.1

Multiply $3$ by $- 1$ .

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

Step 4.2.4.1.2

Simplify $- 3 \ln (3)$ by moving $3$ inside the logarithm.

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

Step 4.2.4.2

Raise $3$ to the power of $3$ .

$f (3) = \frac{2 e - 3 - \ln (27)}{3}$

Step 4.2.5

The final answer is $\frac{2 e - 3 - \ln (27)}{3}$ .

$\frac{2 e - 3 - \ln (27)}{3}$

Step 4.3

Convert $\frac{2 e - 3 - \ln (27)}{3}$ to decimal.

$y = - 0.2864244$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 4.43656365), (2, 1.02513464), (3, - 0.2864244)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 4.437 \\ 2 & 1.025 \\ 3 & - 0.286 \end{array}$

Step 6