Calculus Examples

$y = x e^{2 x}$

Step 1

Find where the expression $x e^{2 x}$ 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 $lim_{x \to - \infty} x e^{2 x}$ to find the horizontal asymptote.

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Step 3.1

Rewrite $x e^{2 x}$ as $\frac{x}{e^{- 2 x}}$ .

$lim_{x \to - \infty} \frac{x}{e^{- 2 x}}$

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.

$\frac{lim_{x \to - \infty} x}{lim_{x \to - \infty} e^{- 2 x}}$

Step 3.2.1.2

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

$\frac{- \infty}{lim_{x \to - \infty} e^{- 2 x}}$

Step 3.2.1.3

Since the exponent $- 2 x$ approaches $\infty$ , the quantity $e^{- 2 x}$ approaches $\infty$ .

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

Step 3.2.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 3.2.2

Since $\frac{- \infty}{\infty}$ 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.

$lim_{x \to - \infty} \frac{x}{e^{- 2 x}} = lim_{x \to - \infty} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [e^{- 2 x}]}$

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.

$lim_{x \to - \infty} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [e^{- 2 x}]}$

Step 3.2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to - \infty} \frac{1}{\frac{d}{d x} [e^{- 2 x}]}$

Step 3.2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - 2 x$ .

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Step 3.2.3.3.1

To apply the Chain Rule, set $u$ as $- 2 x$ .

$lim_{x \to - \infty} \frac{1}{\frac{d}{d u} [e^{u}] \frac{d}{d x} [- 2 x]}$

Step 3.2.3.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$lim_{x \to - \infty} \frac{1}{e^{u} \frac{d}{d x} [- 2 x]}$

Step 3.2.3.3.3

Replace all occurrences of $u$ with $- 2 x$ .

$lim_{x \to - \infty} \frac{1}{e^{- 2 x} \frac{d}{d x} [- 2 x]}$

Step 3.2.3.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$lim_{x \to - \infty} \frac{1}{e^{- 2 x} (- 2 \frac{d}{d x} [x])}$

Step 3.2.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to - \infty} \frac{1}{e^{- 2 x} (- 2 \cdot 1)}$

Step 3.2.3.6

Multiply $- 2$ by $1$ .

$lim_{x \to - \infty} \frac{1}{e^{- 2 x} \cdot - 2}$

Step 3.2.3.7

Move $- 2$ to the left of $e^{- 2 x}$ .

$lim_{x \to - \infty} \frac{1}{- 2 e^{- 2 x}}$

Step 3.2.4

Move the negative in front of the fraction.

$lim_{x \to - \infty} - \frac{1}{2 e^{- 2 x}}$

Step 3.3

Evaluate the limit.

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Step 3.3.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$- lim_{x \to - \infty} \frac{1}{2 e^{- 2 x}}$

Step 3.3.2

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$- \frac{1}{2} lim_{x \to - \infty} \frac{1}{e^{- 2 x}}$

Step 3.4

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{e^{- 2 x}}$ approaches $0$ .

$- \frac{1}{2} \cdot 0$

Step 3.5

Multiply $- \frac{1}{2} \cdot 0$ .

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Step 3.5.1

Multiply $0$ by $- 1$ .

$0 (\frac{1}{2})$

Step 3.5.2

Multiply $0$ by $\frac{1}{2}$ .

$0$

Step 4

List the horizontal asymptotes:

$y = 0$

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: $y = 0$

No Oblique Asymptotes

Step 7