Calculus Examples

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

Step 1

Find the asymptotes.

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Step 1.1

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

$x \leq 0, x = 1$

Step 1.2

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

$x = 1$

Step 1.3

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{\ln (x)}$ approaches $0$ .

$0$

Step 1.4

List the horizontal asymptotes:

$y = 0$

Step 1.5

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

Step 1.6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 1$

Horizontal Asymptotes: $y = 0$

Vertical Asymptotes: $x = 1$

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.

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

Step 2.2

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

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

Step 2.3

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

$y = 1.44269504$

Step 3

Find the point at $x = 3$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$y = 0.91023922$

Step 4

Find the point at $x = 4$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

$y = 0.72134752$

Step 5

The log function can be graphed using the vertical asymptote at $x = 1$ and the points $(2, 1.44269504), (3, 0.91023922), (4, 0.72134752)$ .

Vertical Asymptote: $x = 1$

$\begin{array}{c} x & y \\ 2 & 1.443 \\ 3 & 0.91 \\ 4 & 0.721 \end{array}$

Step 6