Trigonometry Examples

$y = \frac{\ln ({(x)}^{2})}{x}$

Step 1

Find the asymptotes.

Tap for more steps...

Step 1.1

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

$x = 0$

Step 1.2

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 1.3

Find $n$ and $m$ .

$n = 0$

$m = 1$

Step 1.4

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

$y = 0$

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: $x = 0$

Horizontal Asymptotes: $y = 0$

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 0$

Step 2

Find the point at $x = 1$ .

Tap for more steps...

Step 2.1

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

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

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Divide $\ln ({(1)}^{2})$ by $1$ .

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

Step 2.2.2

One to any power is one.

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

Step 2.2.3

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

$f (1) = 0$

Step 2.2.4

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 2$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Expand $\ln ({(2)}^{2})$ by moving $2$ outside the logarithm.

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

Step 3.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

Divide $\ln (2)$ by $1$ .

$f (2) = \ln (2)$

Step 3.2.3

The final answer is $\ln (2)$ .

$\ln (2)$

Step 3.3

Convert $\ln (2)$ to decimal.

$y = 0.69314718$

Step 4

Find the point at $x = 3$ .

Tap for more steps...

Step 4.1

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

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

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Rewrite $\frac{\ln ({(3)}^{2})}{3}$ as $\frac{1}{3} \ln ({(3)}^{2})$ .

$f (3) = \frac{1}{3} \cdot \ln ({(3)}^{2})$

Step 4.2.2

Simplify $\frac{1}{3} \ln ({(3)}^{2})$ by moving $\frac{1}{3}$ inside the logarithm.

$f (3) = \ln ({({(3)}^{2})}^{\frac{1}{3}})$

Step 4.2.3

Multiply the exponents in ${({(3)}^{2})}^{\frac{1}{3}}$ .

Tap for more steps...

Step 4.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (3) = \ln (3^{2 (\frac{1}{3})})$

Step 4.2.3.2

Combine $2$ and $\frac{1}{3}$ .

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

Step 4.2.4

The final answer is $\ln (3^{\frac{2}{3}})$ .

$\ln (3^{\frac{2}{3}})$

Step 4.3

Convert $\ln (3^{\frac{2}{3}})$ to decimal.

$y = 0.73240819$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0), (2, 0.69314718), (3, 0.73240819)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 0.693 \\ 3 & 0.732 \end{array}$

Step 6