Calculus Examples

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

Step 1

Find the asymptotes.

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Step 1.1

Find where the expression $\frac{1}{x \ln (x)}$ is undefined.

$x \leq 0, x = 1$

Step 1.2

Since $\frac{1}{x \ln (x)}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the left and $\frac{1}{x \ln (x)}$ $\to$ $- \infty$ as $x$ $\to$ $0$ from the right, then $x = 0$ is a vertical asymptote.

$x = 0$

Step 1.3

Since $\frac{1}{x \ln (x)}$ $\to$ $- \infty$ as $x$ $\to$ $1$ from the left and $\frac{1}{x \ln (x)}$ $\to$ $\infty$ as $x$ $\to$ $1$ from the right, then $x = 1$ is a vertical asymptote.

$x = 1$

Step 1.4

List all of the vertical asymptotes:

$x = 0, 1$

Step 1.5

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 1.6

Find $n$ and $m$ .

$n = 0$

$m = 1$

Step 1.7

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

$y = 0$

Step 1.8

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

Step 1.9

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0, 1$

Horizontal Asymptotes: $y = 0$

Vertical Asymptotes: $x = 0, 1$

Horizontal Asymptotes: $y = 0$

Step 2

Find the point at $x = 2$ .

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Step 2.1

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify $2 \ln (2)$ by moving $2$ inside the logarithm.

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

Step 2.2.2

Raise $2$ to the power of $2$ .

$f (2) = \frac{1}{\ln (4)}$

Step 2.2.3

The final answer is $\frac{1}{\ln (4)}$ .

$\frac{1}{\ln (4)}$

Step 2.3

Convert $\frac{1}{\ln (4)}$ to decimal.

$y = 0.72134752$

Step 3

Find the point at $x = 3$ .

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

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

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

Step 3.2.2

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

$f (3) = \frac{1}{\ln (27)}$

Step 3.2.3

The final answer is $\frac{1}{\ln (27)}$ .

$\frac{1}{\ln (27)}$

Step 3.3

Convert $\frac{1}{\ln (27)}$ to decimal.

$y = 0.30341307$

Step 4

The log function can be graphed using the vertical asymptote at $x = 0, 1$ and the points $(2, 0.72134752), (3, 0.30341307)$ .

Vertical Asymptote: $x = 0, 1$

$\begin{array}{c} x & y \\ 2 & 0.721 \\ 3 & 0.303 \end{array}$

Step 5