Calculus Examples

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

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

Step 1

Find the asymptotes.

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Step 1.1

Find where the expression

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

$\frac{\ln (x)}{x}$ is undefined.

x \leq 0

$x \leq 0$

Step 1.2

Since

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

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

\to

$\to$

\infty

$\infty$ as

x

$x$

\to

$\to$

0

$0$ from the left and

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

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

\to

$\to$

- \infty

$- \infty$ as

x

$x$

\to

$\to$

0

$0$ from the right, then

x = 0

$x = 0$ is a vertical asymptote.

x = 0

$x = 0$

Step 1.3

Ignoring the logarithm, consider the rational function

R (x) = \frac{a x^{n}}{b x^{m}}

$R (x) = \frac{a x^{n}}{b x^{m}}$ where

n

$n$ is the degree of the numerator and

m

$m$ is the degree of the denominator.

1. If

n < m

$n < m$ , then the x-axis,

y = 0

$y = 0$ , is the horizontal asymptote.

2. If

n = m

$n = m$ , then the horizontal asymptote is the line

y = \frac{a}{b}

$y = \frac{a}{b}$ .

3. If

n > m

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

Step 1.4

Find

n

$n$ and

m

$m$ .

n = 0

$n = 0$

m = 1

$m = 1$

Step 1.5

Since

n < m

$n < m$ , the x-axis,

y = 0

$y = 0$ , is the horizontal asymptote.

y = 0

$y = 0$

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

$x = 0$

Horizontal Asymptotes:

y = 0

$y = 0$

Vertical Asymptotes:

x = 0

$x = 0$

Horizontal Asymptotes:

y = 0

$y = 0$

Step 2

Find the point at

x = 1

$x = 1$ .

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Step 2.1

Replace the variable

x

$x$ with

1

$1$ in the expression.

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Divide

\ln (1)

$\ln (1)$ by

1

$1$ .

f (1) = \ln (1)

$f (1) = \ln (1)$

Step 2.2.2

The natural logarithm of

1

$1$ is

0

$0$ .

f (1) = 0

$f (1) = 0$

Step 2.2.3

The final answer is

0

$0$ .

0

$0$

0

$0$

Step 2.3

Convert

0

$0$ to decimal.

y = 0

$y = 0$

y = 0

$y = 0$

Step 3

Find the point at

x = 2

$x = 2$ .

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Step 3.1

Replace the variable

x

$x$ with

2

$2$ in the expression.

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

Rewrite

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

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

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

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

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

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

Step 3.2.2

Simplify

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

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

\frac{1}{2}

$\frac{1}{2}$ inside the logarithm.

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

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

Step 3.2.3

The final answer is

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

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

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

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

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

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

Step 3.3

Convert

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

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

y = 0.34657359

$y = 0.34657359$

y = 0.34657359

$y = 0.34657359$

Step 4

Find the point at

x = 3

$x = 3$ .

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Step 4.1

Replace the variable

x

$x$ with

3

$3$ in the expression.

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Rewrite

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

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

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

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

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

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

Step 4.2.2

Simplify

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

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

\frac{1}{3}

$\frac{1}{3}$ inside the logarithm.

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

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

Step 4.2.3

The final answer is

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

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

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

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

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

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

Step 4.3

Convert

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

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

y = 0.36620409

$y = 0.36620409$

y = 0.36620409

$y = 0.36620409$

Step 5

The log function can be graphed using the vertical asymptote at

x = 0

$x = 0$ and the points

(1, 0), (2, 0.34657359), (3, 0.36620409)

$(1, 0), (2, 0.34657359), (3, 0.36620409)$ .

Vertical Asymptote:

x = 0

$x = 0$

\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 0.347 \\ 3 & 0.366 \end{array}

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 0.347 \\ 3 & 0.366 \end{array}$

Step 6