Trigonometry Examples

x e^{- x}

$x e^{- x}$

Step 1

Find where the expression

x e^{- x}

$x e^{- x}$ is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

Evaluate

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

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

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Step 3.1

Rewrite

x e^{- x}

$x e^{- x}$ as

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

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

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

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

Step 3.2

Apply L'Hospital's rule.

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Step 3.2.1

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

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Step 3.2.1.1

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

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

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

Step 3.2.1.2

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

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

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

Step 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 3.2.1.4

Infinity divided by infinity is undefined.

Undefined

\frac{\infty}{\infty}

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

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

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

Step 3.2.3

Find the derivative of the numerator and denominator.

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Step 3.2.3.1

Differentiate the numerator and denominator.

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

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

Step 3.2.3.2

Differentiate using the Power Rule which states that

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

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

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

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

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

Step 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{1}{e^{x}}

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

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

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

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

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

Step 3.3

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

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

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

0

$0$ .

0

$0$

0

$0$

Step 4

List the horizontal asymptotes:

y = 0

$y = 0$

Step 5

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

Step 6

This is the set of all asymptotes.

No Vertical Asymptotes

Horizontal Asymptotes:

y = 0

$y = 0$

No Oblique Asymptotes

Step 7