Calculus Examples

$lim_{x \to \infty} \frac{\ln (3 x)}{e^{7 x}}$

Step 1

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

Tap for more steps...

Step 1.1

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

$\frac{lim_{x \to \infty} \ln (3 x)}{lim_{x \to \infty} e^{7 x}}$

Step 1.2

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

$\frac{\infty}{lim_{x \to \infty} e^{7 x}}$

Step 1.3

Since the exponent $7 x$ approaches $\infty$ , the quantity $e^{7 x}$ approaches $\infty$ .

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

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.

$lim_{x \to \infty} \frac{\ln (3 x)}{e^{7 x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (3 x)]}{\frac{d}{d x} [e^{7 x}]}$

Step 3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (3 x)]}{\frac{d}{d x} [e^{7 x}]}$

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 $f (x) = \ln (x)$ and $g (x) = 3 x$ .

Tap for more steps...

Step 3.2.1

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

$lim_{x \to \infty} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 x]}{\frac{d}{d x} [e^{7 x}]}$

Step 3.2.2

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

$lim_{x \to \infty} \frac{\frac{1}{u} \frac{d}{d x} [3 x]}{\frac{d}{d x} [e^{7 x}]}$

Step 3.2.3

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

$lim_{x \to \infty} \frac{\frac{1}{3 x} \frac{d}{d x} [3 x]}{\frac{d}{d x} [e^{7 x}]}$

Step 3.3

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

$lim_{x \to \infty} \frac{\frac{1}{3 x} (3 \frac{d}{d x} [x])}{\frac{d}{d x} [e^{7 x}]}$

Step 3.4

Combine $3$ and $\frac{1}{3 x}$ .

$lim_{x \to \infty} \frac{\frac{3}{3 x} \frac{d}{d x} [x]}{\frac{d}{d x} [e^{7 x}]}$

Step 3.5

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.5.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{3}{3 x} \frac{d}{d x} [x]}{\frac{d}{d x} [e^{7 x}]}$

Step 3.5.2

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{1}{x} \frac{d}{d x} [x]}{\frac{d}{d x} [e^{7 x}]}$

Step 3.6

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

$lim_{x \to \infty} \frac{\frac{1}{x} \cdot 1}{\frac{d}{d x} [e^{7 x}]}$

Step 3.7

Multiply $\frac{1}{x}$ by $1$ .

$lim_{x \to \infty} \frac{\frac{1}{x}}{\frac{d}{d x} [e^{7 x}]}$

Step 3.8

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 $g (x) = 7 x$ .

Tap for more steps...

Step 3.8.1

To apply the Chain Rule, set $u$ as $7 x$ .

$lim_{x \to \infty} \frac{\frac{1}{x}}{\frac{d}{d u} [e^{u}] \frac{d}{d x} [7 x]}$

Step 3.8.2

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

$lim_{x \to \infty} \frac{\frac{1}{x}}{e^{u} \frac{d}{d x} [7 x]}$

Step 3.8.3

Replace all occurrences of $u$ with $7 x$ .

$lim_{x \to \infty} \frac{\frac{1}{x}}{e^{7 x} \frac{d}{d x} [7 x]}$

Step 3.9

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$lim_{x \to \infty} \frac{\frac{1}{x}}{e^{7 x} (7 \frac{d}{d x} [x])}$

Step 3.10

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

$lim_{x \to \infty} \frac{\frac{1}{x}}{e^{7 x} (7 \cdot 1)}$

Step 3.11

Multiply $7$ by $1$ .

$lim_{x \to \infty} \frac{\frac{1}{x}}{e^{7 x} \cdot 7}$

Step 3.12

Move $7$ to the left of $e^{7 x}$ .

$lim_{x \to \infty} \frac{\frac{1}{x}}{7 \cdot e^{7 x}}$

Step 3.13

Multiply $7$ by $e^{7 x}$ .

$lim_{x \to \infty} \frac{\frac{1}{x}}{7 e^{7 x}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to \infty} \frac{1}{x} \cdot \frac{1}{7 e^{7 x}}$

Step 5

Evaluate the limit.

Tap for more steps...

Step 5.1

Multiply $\frac{1}{x}$ by $\frac{1}{7 e^{7 x}}$ .

$lim_{x \to \infty} \frac{1}{x (7 e^{7 x})}$

Step 5.2

Move the term $\frac{1}{7}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{7} lim_{x \to \infty} \frac{1}{x (e^{7 x})}$

Step 6

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x (e^{7 x})}$ approaches $0$ .

$\frac{1}{7} \cdot 0$

Step 7

Multiply $\frac{1}{7}$ by $0$ .

$0$