Calculus Examples

$\frac{\ln (x)}{7 x}$

Step 1

Find the asymptotes.

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Find where the expression $\frac{\ln (x)}{7 x}$ is undefined.

$x \leq 0$

Since $\frac{\ln (x)}{7 x}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the left and $\frac{\ln (x)}{7 x}$ $\to$ $- \infty$ as $x$ $\to$ $0$ from the right, then $x = 0$ is a vertical asymptote.

$x = 0$

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).

Find $n$ and $m$ .

$n = 0$

$m = 1$

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

$y = 0$

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 0$

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 0$

Step 2

Find the point at $x = 1$ .

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

$f (1) = \frac{\ln (1)}{7 (1)}$

Simplify the result.

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

$f (1) = \frac{0}{7 (1)}$

Multiply $7$ by $1$ .

$f (1) = \frac{0}{7}$

Divide $0$ by $7$ .

$f (1) = 0$

The final answer is $0$ .

$0$

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 2$ .

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

$f (2) = \frac{\ln (2)}{7 (2)}$

Simplify the result.

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Multiply $7$ by $2$ .

$f (2) = \frac{\ln (2)}{14}$

Rewrite $\frac{\ln (2)}{14}$ as $\frac{1}{14} \ln (2)$ .

$f (2) = \frac{1}{14} \cdot \ln (2)$

Simplify $\frac{1}{14} \ln (2)$ by moving $\frac{1}{14}$ inside the logarithm.

$f (2) = \ln (2^{\frac{1}{14}})$

The final answer is $\ln (2^{\frac{1}{14}})$ .

$\ln (2^{\frac{1}{14}})$

Convert $\ln (2^{\frac{1}{14}})$ to decimal.

$y = 0.04951051$

Step 4

Find the point at $x = 3$ .

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

$f (3) = \frac{\ln (3)}{7 (3)}$

Simplify the result.

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Multiply $7$ by $3$ .

$f (3) = \frac{\ln (3)}{21}$

Rewrite $\frac{\ln (3)}{21}$ as $\frac{1}{21} \ln (3)$ .

$f (3) = \frac{1}{21} \cdot \ln (3)$

Simplify $\frac{1}{21} \ln (3)$ by moving $\frac{1}{21}$ inside the logarithm.

$f (3) = \ln (3^{\frac{1}{21}})$

The final answer is $\ln (3^{\frac{1}{21}})$ .

$\ln (3^{\frac{1}{21}})$

Convert $\ln (3^{\frac{1}{21}})$ to decimal.

$y = 0.05231487$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0), (2, 0.04951051), (3, 0.05231487)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 0.05 \\ 3 & 0.052 \end{array}$

Step 6