Calculus Examples

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

Step 1

Find the asymptotes.

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Step 1.1

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

$x \leq 0$

Step 1.2

Since $\frac{\ln (x)}{x^{2}}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the left and $\frac{\ln (x)}{x^{2}}$ $\to$ $- \infty$ as $x$ $\to$ $0$ from the right, then $x = 0$ is a vertical asymptote.

$x = 0$

Step 1.3

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 1.4

Find $n$ and $m$ .

$n = 0$

$m = 2$

Step 1.5

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

$y = 0$

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: $x = 0$

Horizontal Asymptotes: $y = 0$

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 0$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

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

$f (1) = \frac{0}{1^{2}}$

Step 2.2.2

One to any power is one.

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

Step 2.2.3

Divide $0$ by $1$ .

$f (1) = 0$

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 $x = 2$ .

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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.

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Step 3.2.1

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

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

Step 3.2.2

Rewrite $\frac{\ln (2)}{4}$ as $\frac{1}{4} \ln (2)$ .

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

Step 3.2.3

Simplify $\frac{1}{4} \ln (2)$ by moving $\frac{1}{4}$ inside the logarithm.

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

Step 3.2.4

The final answer is $\ln (2^{\frac{1}{4}})$ .

$\ln (2^{\frac{1}{4}})$

Step 3.3

Convert $\ln (2^{\frac{1}{4}})$ to decimal.

$y = 0.17328679$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Raise $3$ to the power of $2$ .

$f (3) = \frac{\ln (3)}{9}$

Step 4.2.2

Rewrite $\frac{\ln (3)}{9}$ as $\frac{1}{9} \ln (3)$ .

$f (3) = \frac{1}{9} \cdot \ln (3)$

Step 4.2.3

Simplify $\frac{1}{9} \ln (3)$ by moving $\frac{1}{9}$ inside the logarithm.

$f (3) = \ln (3^{\frac{1}{9}})$

Step 4.2.4

The final answer is $\ln (3^{\frac{1}{9}})$ .

$\ln (3^{\frac{1}{9}})$

Step 4.3

Convert $\ln (3^{\frac{1}{9}})$ to decimal.

$y = 0.12206803$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0), (2, 0.17328679), (3, 0.12206803)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 0.173 \\ 3 & 0.122 \end{array}$

Step 6