Calculus Examples

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

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.

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

Step 1.2

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

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

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

Step 6

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

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

Step 7

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

$\infty \cdot \infty$

Step 8

Infinity times infinity is infinity.

$\infty$