Calculus Examples

$lim_{x \to \infty} \frac{e^{\frac{x}{18}}}{x}$

Step 1

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

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Step 1.1

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

$\frac{lim_{x \to \infty} e^{\frac{x}{18}}}{lim_{x \to \infty} x}$

Step 1.2

Since the exponent $\frac{x}{18}$ approaches $\infty$ , the quantity $e^{\frac{x}{18}}$ approaches $\infty$ .

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

Step 1.3

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

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

Step 1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 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.

$lim_{x \to \infty} \frac{e^{\frac{x}{18}}}{x} = lim_{x \to \infty} \frac{\frac{d}{d x} [e^{\frac{x}{18}}]}{\frac{d}{d x} [x]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [e^{\frac{x}{18}}]}{\frac{d}{d x} [x]}$

Step 3.2

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 $g (x) = \frac{x}{18}$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $\frac{x}{18}$ .

$lim_{x \to \infty} \frac{\frac{d}{d u} [e^{u}] \frac{d}{d x} [\frac{x}{18}]}{\frac{d}{d x} [x]}$

Step 3.2.2

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

$lim_{x \to \infty} \frac{e^{u} \frac{d}{d x} [\frac{x}{18}]}{\frac{d}{d x} [x]}$

Step 3.2.3

Replace all occurrences of $u$ with $\frac{x}{18}$ .

$lim_{x \to \infty} \frac{e^{\frac{x}{18}} \frac{d}{d x} [\frac{x}{18}]}{\frac{d}{d x} [x]}$

Step 3.3

Since $\frac{1}{18}$ is constant with respect to $x$ , the derivative of $\frac{x}{18}$ with respect to $x$ is $\frac{1}{18} \frac{d}{d x} [x]$ .

$lim_{x \to \infty} \frac{e^{\frac{x}{18}} (\frac{1}{18} \frac{d}{d x} [x])}{\frac{d}{d x} [x]}$

Step 3.4

Combine $\frac{1}{18}$ and $e^{\frac{x}{18}}$ .

$lim_{x \to \infty} \frac{\frac{e^{\frac{x}{18}}}{18} \frac{d}{d x} [x]}{\frac{d}{d x} [x]}$

Step 3.5

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

$lim_{x \to \infty} \frac{\frac{e^{\frac{x}{18}}}{18} \cdot 1}{\frac{d}{d x} [x]}$

Step 3.6

Multiply $\frac{e^{\frac{x}{18}}}{18}$ by $1$ .

$lim_{x \to \infty} \frac{\frac{e^{\frac{x}{18}}}{18}}{\frac{d}{d x} [x]}$

Step 3.7

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

$lim_{x \to \infty} \frac{\frac{e^{\frac{x}{18}}}{18}}{1}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to \infty} \frac{e^{\frac{x}{18}}}{18} \cdot 1$

Step 5

Multiply $\frac{e^{\frac{x}{18}}}{18}$ by $1$ .

$lim_{x \to \infty} \frac{e^{\frac{x}{18}}}{18}$

Step 6

Since the function $e^{\frac{1}{18} x}$ approaches $\infty$ , the positive constant $\frac{1}{18}$ times the function also approaches $\infty$ .

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Step 6.1

Consider the limit with the constant multiple $\frac{1}{18}$ removed.

$lim_{x \to \infty} e^{\frac{1}{18} x}$

Step 6.2

Since the exponent $\frac{1}{18} x$ approaches $\infty$ , the quantity $e^{\frac{1}{18} x}$ approaches $\infty$ .

$\infty$