Calculus Examples

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

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.

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

Step 1.2

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

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

Step 1.3

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

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $3$ .

$lim_{x \to \infty} \frac{1}{3^{x} \ln (3)}$

Step 4

Move the term $\frac{1}{\ln (3)}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{\ln (3)} lim_{x \to \infty} \frac{1}{3^{x}}$

Step 5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{3^{x}}$ approaches $0$ .

$\frac{1}{\ln (3)} \cdot 0$

Step 6

Multiply $\frac{1}{\ln (3)}$ by $0$ .

$0$