Algebra Examples

$A (t) = 300 t e^{- 2.5 t}$

Step 1

Find where the expression $300 x e^{- 2.5 x}$ 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 $lim_{x \to \infty} 300 x e^{- 2.5 x}$ to find the horizontal asymptote.

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Step 3.1

Move the term $300$ outside of the limit because it is constant with respect to $x$ .

$300 lim_{x \to \infty} x e^{- 2.5 x}$

Step 3.2

Rewrite $x e^{- 2.5 x}$ as $\frac{x}{e^{2.5 x}}$ .

$300 lim_{x \to \infty} \frac{x}{e^{2.5 x}}$

Step 3.3

Apply L'Hospital's rule.

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Step 3.3.1

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

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Step 3.3.1.1

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

$300 \frac{lim_{x \to \infty} x}{lim_{x \to \infty} e^{2.5 x}}$

Step 3.3.1.2

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

$300 \frac{\infty}{lim_{x \to \infty} e^{2.5 x}}$

Step 3.3.1.3

Since the exponent $2.5 x$ approaches $\infty$ , the quantity $e^{2.5 x}$ approaches $\infty$ .

$300 \frac{\infty}{\infty}$

Step 3.3.1.4

Infinity divided by infinity is undefined.

Undefined

$300 \frac{\infty}{\infty}$

Step 3.3.2

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.

$lim_{x \to \infty} \frac{x}{e^{2.5 x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [e^{2.5 x}]}$

Step 3.3.3

Find the derivative of the numerator and denominator.

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Step 3.3.3.1

Differentiate the numerator and denominator.

$300 lim_{x \to \infty} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [e^{2.5 x}]}$

Step 3.3.3.2

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

$300 lim_{x \to \infty} \frac{1}{\frac{d}{d x} [e^{2.5 x}]}$

Step 3.3.3.3

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 $g (x) = 2.5 x$ .

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Step 3.3.3.3.1

To apply the Chain Rule, set $u$ as $2.5 x$ .

$300 lim_{x \to \infty} \frac{1}{\frac{d}{d u} [e^{u}] \frac{d}{d x} [2.5 x]}$

Step 3.3.3.3.2

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

$300 lim_{x \to \infty} \frac{1}{e^{u} \frac{d}{d x} [2.5 x]}$

Step 3.3.3.3.3

Replace all occurrences of $u$ with $2.5 x$ .

$300 lim_{x \to \infty} \frac{1}{e^{2.5 x} \frac{d}{d x} [2.5 x]}$

Step 3.3.3.4

Since $2.5$ is constant with respect to $x$ , the derivative of $2.5 x$ with respect to $x$ is $2.5 \frac{d}{d x} [x]$ .

$300 lim_{x \to \infty} \frac{1}{e^{2.5 x} \cdot 2.5 \frac{d}{d x} [x]}$

Step 3.3.3.5

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

$300 lim_{x \to \infty} \frac{1}{e^{2.5 x} \cdot 2.5 \cdot 1}$

Step 3.3.3.6

Multiply $2.5$ by $1$ .

$300 lim_{x \to \infty} \frac{1}{e^{2.5 x} \cdot 2.5}$

Step 3.3.3.7

Move $2.5$ to the left of $e^{2.5 x}$ .

$300 lim_{x \to \infty} \frac{1}{2.5 e^{2.5 x}}$

Step 3.3.4

Rewrite $1$ as $1 (1)$ .

$300 lim_{x \to \infty} \frac{1 (1)}{2.5 e^{2.5 x}}$

Step 3.3.5

Factor $2.5$ out of $2.5 e^{2.5 x}$ .

$300 lim_{x \to \infty} \frac{1 (1)}{2.5 (e^{2.5 x})}$

Step 3.3.6

Separate fractions.

$300 lim_{x \to \infty} \frac{1}{2.5} \cdot \frac{1}{e^{2.5 x}}$

Step 3.3.7

Divide $1$ by $2.5$ .

$300 lim_{x \to \infty} 0.4 \frac{1}{e^{2.5 x}}$

Step 3.3.8

Combine $0.4$ and $\frac{1}{e^{2.5 x}}$ .

$300 lim_{x \to \infty} \frac{0.4}{e^{2.5 x}}$

Step 3.4

Move the term $0.4$ outside of the limit because it is constant with respect to $x$ .

$300 \cdot 0.4 lim_{x \to \infty} \frac{1}{e^{2.5 x}}$

Step 3.5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{e^{2.5 x}}$ approaches $0$ .

$300 \cdot 0.4 \cdot 0$

Step 3.6

Multiply by zero.

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Step 3.6.1

Multiply $0.4$ by $0$ .

$300 \cdot 0$

Step 3.6.2

Multiply $300$ by $0$ .

$0$

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