Calculus Examples

$lim_{x \to \infty} \frac{e^{3 x}}{\ln (x)}$

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.

$\frac{lim_{x \to \infty} e^{3 x}}{lim_{x \to \infty} \ln (x)}$

Step 1.2

Since the exponent $3 x$ approaches $\infty$ , the quantity $e^{3 x}$ approaches $\infty$ .

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

Step 1.3

As log approaches infinity, the value goes to $\infty$ .

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

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.

$lim_{x \to \infty} \frac{e^{3 x}}{\ln (x)} = lim_{x \to \infty} \frac{\frac{d}{d x} [e^{3 x}]}{\frac{d}{d x} [\ln (x)]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [e^{3 x}]}{\frac{d}{d x} [\ln (x)]}$

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 $g (x) = 3 x$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $3 x$ .

$lim_{x \to \infty} \frac{\frac{d}{d u} [e^{u}] \frac{d}{d x} [3 x]}{\frac{d}{d x} [\ln (x)]}$

Step 3.2.2

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

$lim_{x \to \infty} \frac{e^{u} \frac{d}{d x} [3 x]}{\frac{d}{d x} [\ln (x)]}$

Step 3.2.3

Replace all occurrences of $u$ with $3 x$ .

$lim_{x \to \infty} \frac{e^{3 x} \frac{d}{d x} [3 x]}{\frac{d}{d x} [\ln (x)]}$

Step 3.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$lim_{x \to \infty} \frac{e^{3 x} (3 \frac{d}{d x} [x])}{\frac{d}{d x} [\ln (x)]}$

Step 3.4

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

$lim_{x \to \infty} \frac{e^{3 x} (3 \cdot 1)}{\frac{d}{d x} [\ln (x)]}$

Step 3.5

Multiply $3$ by $1$ .

$lim_{x \to \infty} \frac{e^{3 x} \cdot 3}{\frac{d}{d x} [\ln (x)]}$

Step 3.6

Move $3$ to the left of $e^{3 x}$ .

$lim_{x \to \infty} \frac{3 \cdot e^{3 x}}{\frac{d}{d x} [\ln (x)]}$

Step 3.7

Multiply $3$ by $e^{3 x}$ .

$lim_{x \to \infty} \frac{3 e^{3 x}}{\frac{d}{d x} [\ln (x)]}$

Step 3.8

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$lim_{x \to \infty} \frac{3 e^{3 x}}{\frac{1}{x}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to \infty} 3 e^{3 x} x$

Step 5

Evaluate the limit.

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Step 5.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $\infty$ .

$lim_{x \to \infty} 3 \cdot lim_{x \to \infty} e^{3 x} \cdot lim_{x \to \infty} x$

Step 5.2

Evaluate the limit of $3$ which is constant as $x$ approaches $\infty$ .

$3 \cdot lim_{x \to \infty} e^{3 x} \cdot lim_{x \to \infty} x$

Step 6

Since the exponent $3 x$ approaches $\infty$ , the quantity $e^{3 x}$ approaches $\infty$ .

$3 \cdot \infty \cdot lim_{x \to \infty} x$

Step 7

Evaluate the limit.

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Step 7.1

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

$3 \cdot \infty \cdot \infty$

Step 7.2

Simplify the answer.

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Step 7.2.1

A non-zero constant times infinity is infinity.

$\infty \cdot \infty$

Step 7.2.2

Infinity times infinity is infinity.

$\infty$