Finite Math Examples

y = e^{- x} \cdot ln (x)

$y = e^{- x} \cdot \ln (x)$

Step 1

Find the asymptotes.

Tap for more steps...

Step 1.1

Find where the expression

e^{- x} \cdot ln (x)

$e^{- x} \cdot \ln (x)$ is undefined.

x \leq 0

$x \leq 0$

Step 1.2

Since

e^{- x} \cdot ln (x)

$e^{- x} \cdot \ln (x)$

\to

$\to$

\infty

$\infty$ as

x

$x$

\to

$\to$

0

$0$ from the left and

e^{- x} \cdot ln (x)

$e^{- x} \cdot \ln (x)$

\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

Evaluate

lim x \to \infty e^{- x} ln (x)

$lim_{x \to \infty} e^{- x} \ln (x)$ to find the horizontal asymptote.

Tap for more steps...

Step 1.3.1

Rewrite

e^{- x} ln (x)

$e^{- x} \ln (x)$ as

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

$\frac{\ln (x)}{e^{x}}$ .

lim x \to \infty \frac{ln (x)}{e^{x}}

$lim_{x \to \infty} \frac{\ln (x)}{e^{x}}$

Step 1.3.2

Apply L'Hospital's rule.

Tap for more steps...

Step 1.3.2.1

Evaluate the limit of the numerator and the limit of the denominator.

Tap for more steps...

Step 1.3.2.1.1

Take the limit of the numerator and the limit of the denominator.

\frac{lim x \to \infty ln (x)}{lim x \to \infty e^{x}}

$\frac{lim_{x \to \infty} \ln (x)}{lim_{x \to \infty} e^{x}}$

Step 1.3.2.1.2

As log approaches infinity, the value goes to

\infty

$\infty$ .

\frac{\infty}{lim x \to \infty e^{x}}

$\frac{\infty}{lim_{x \to \infty} e^{x}}$

Step 1.3.2.1.3

Since the exponent

x

$x$ approaches

\infty

$\infty$ , the quantity

e^{x}

$e^{x}$ approaches

\infty

$\infty$ .

\frac{\infty}{\infty}

$\frac{\infty}{\infty}$

Step 1.3.2.1.4

Infinity divided by infinity is undefined.

Undefined

\frac{\infty}{\infty}

$\frac{\infty}{\infty}$

Step 1.3.2.2

Since

\frac{\infty}{\infty}

$\frac{\infty}{\infty}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

lim x \to \infty \frac{ln (x)}{e^{x}} = lim x \to \infty \frac{\frac{d}{d x} [ln (x)]}{\frac{d}{d x} [e^{x}]}

$lim_{x \to \infty} \frac{\ln (x)}{e^{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [e^{x}]}$

Step 1.3.2.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 1.3.2.3.1

Differentiate the numerator and denominator.

lim x \to \infty \frac{\frac{d}{d x} [ln (x)]}{\frac{d}{d x} [e^{x}]}

$lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [e^{x}]}$

Step 1.3.2.3.2

The derivative of

ln (x)

$\ln (x)$ with respect to

x

$x$ is

\frac{1}{x}

$\frac{1}{x}$ .

lim x \to \infty \frac{\frac{1}{x}}{\frac{d}{d x} [e^{x}]}

$lim_{x \to \infty} \frac{\frac{1}{x}}{\frac{d}{d x} [e^{x}]}$

Step 1.3.2.3.3

Differentiate using the Exponential Rule which states that

\frac{d}{d x} [a^{x}]

$\frac{d}{d x} [a^{x}]$ is

a^{x} ln (a)

$a^{x} \ln (a)$ where

a

$a$ =

e

$e$ .

lim x \to \infty \frac{\frac{1}{x}}{e^{x}}

$lim_{x \to \infty} \frac{\frac{1}{x}}{e^{x}}$

lim x \to \infty \frac{\frac{1}{x}}{e^{x}}

$lim_{x \to \infty} \frac{\frac{1}{x}}{e^{x}}$

Step 1.3.2.4

Multiply the numerator by the reciprocal of the denominator.

lim x \to \infty \frac{1}{x} \cdot \frac{1}{e^{x}}

$lim_{x \to \infty} \frac{1}{x} \cdot \frac{1}{e^{x}}$

Step 1.3.2.5

Multiply

\frac{1}{x}

$\frac{1}{x}$ by

\frac{1}{e^{x}}

$\frac{1}{e^{x}}$ .

lim x \to \infty \frac{1}{x e^{x}}

$lim_{x \to \infty} \frac{1}{x e^{x}}$

lim x \to \infty \frac{1}{x e^{x}}

$lim_{x \to \infty} \frac{1}{x e^{x}}$

Step 1.3.3

Since its numerator approaches a real number while its denominator is unbounded, the fraction

\frac{1}{x e^{x}}

$\frac{1}{x e^{x}}$ approaches

0

$0$ .

0

$0$

0

$0$

Step 1.4

List the horizontal asymptotes:

y = 0

$y = 0$

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

$x = 0$

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

$x = 1$ .

Tap for more steps...

Step 2.1

Replace the variable

x

$x$ with

1

$1$ in the expression.

f (1) = e^{- (1)} \cdot ln (1)

$f (1) = e^{- (1)} \cdot \ln (1)$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Multiply

- 1

$- 1$ by

1

$1$ .

f (1) = e^{- 1} \cdot ln (1)

$f (1) = e^{- 1} \cdot \ln (1)$

Step 2.2.2

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 2.2.3

The natural logarithm of

1

$1$ is

0

$0$ .

f (1) = \frac{1}{e} \cdot 0

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

Step 2.2.4

Multiply

\frac{1}{e}

$\frac{1}{e}$ by

0

$0$ .

f (1) = 0

$f (1) = 0$

Step 2.2.5

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

$x = 2$ .

Tap for more steps...

Step 3.1

Replace the variable

x

$x$ with

2

$2$ in the expression.

f (2) = e^{- (2)} \cdot ln (2)

$f (2) = e^{- (2)} \cdot \ln (2)$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Multiply

- 1

$- 1$ by

2

$2$ .

f (2) = e^{- 2} \cdot ln (2)

$f (2) = e^{- 2} \cdot \ln (2)$

Step 3.2.2

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 3.2.3

Combine

\frac{1}{e^{2}}

$\frac{1}{e^{2}}$ and

ln (2)

$\ln (2)$ .

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

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

Step 3.2.4

The final answer is

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

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

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

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

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

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

Step 3.3

Convert

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

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

y = 0.09380727

$y = 0.09380727$

y = 0.09380727

$y = 0.09380727$

Step 4

Find the point at

x = 3

$x = 3$ .

Tap for more steps...

Step 4.1

Replace the variable

x

$x$ with

3

$3$ in the expression.

f (3) = e^{- (3)} \cdot ln (3)

$f (3) = e^{- (3)} \cdot \ln (3)$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Multiply

$- 1$ by

$3$ .

$f (3) = e^{- 3} \cdot \ln (3)$

Step 4.2.2

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.2.3

Combine

$\frac{1}{e^{3}}$ and

$\ln (3)$ .

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

Step 4.2.4

The final answer is

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

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

Step 4.3

Convert

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

$y = 0.05469668$

Step 5

The log function can be graphed using the vertical asymptote at

$x = 0$ and the points

$(1, 0), (2, 0.09380727), (3, 0.05469668)$ .

Vertical Asymptote:

$x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 0.094 \\ 3 & 0.055 \end{array}$

Step 6