Calculus Examples

$lim_{x \to \infty} \frac{\ln (x^{4})}{x^{3}}$

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.

$\frac{lim_{x \to \infty} \ln (x^{4})}{lim_{x \to \infty} x^{3}}$

Step 1.2

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

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

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.

$lim_{x \to \infty} \frac{\ln (x^{4})}{x^{3}} = lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (x^{4})]}{\frac{d}{d x} [x^{3}]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (x^{4})]}{\frac{d}{d x} [x^{3}]}$

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 $g (x) = x^{4}$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $x^{4}$ .

$lim_{x \to \infty} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{4}]}{\frac{d}{d x} [x^{3}]}$

Step 3.2.2

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

$lim_{x \to \infty} \frac{\frac{1}{u} \frac{d}{d x} [x^{4}]}{\frac{d}{d x} [x^{3}]}$

Step 3.2.3

Replace all occurrences of $u$ with $x^{4}$ .

$lim_{x \to \infty} \frac{\frac{1}{x^{4}} \frac{d}{d x} [x^{4}]}{\frac{d}{d x} [x^{3}]}$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$lim_{x \to \infty} \frac{\frac{1}{x^{4}} (4 x^{3})}{\frac{d}{d x} [x^{3}]}$

Step 3.4

Combine $4$ and $\frac{1}{x^{4}}$ .

$lim_{x \to \infty} \frac{\frac{4}{x^{4}} x^{3}}{\frac{d}{d x} [x^{3}]}$

Step 3.5

Combine $\frac{4}{x^{4}}$ and $x^{3}$ .

$lim_{x \to \infty} \frac{\frac{4 x^{3}}{x^{4}}}{\frac{d}{d x} [x^{3}]}$

Step 3.6

Cancel the common factor of $x^{3}$ and $x^{4}$ .

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Step 3.6.1

Factor $x^{3}$ out of $4 x^{3}$ .

$lim_{x \to \infty} \frac{\frac{x^{3} \cdot 4}{x^{4}}}{\frac{d}{d x} [x^{3}]}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $x^{3}$ out of $x^{4}$ .

$lim_{x \to \infty} \frac{\frac{x^{3} \cdot 4}{x^{3} x}}{\frac{d}{d x} [x^{3}]}$

Step 3.6.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{x^{3} \cdot 4}{x^{3} x}}{\frac{d}{d x} [x^{3}]}$

Step 3.6.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{4}{x}}{\frac{d}{d x} [x^{3}]}$

Step 3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$lim_{x \to \infty} \frac{\frac{4}{x}}{3 x^{2}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to \infty} \frac{4}{x} \cdot \frac{1}{3 x^{2}}$

Step 5

Combine factors.

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Step 5.1

Multiply $\frac{4}{x}$ by $\frac{1}{3 x^{2}}$ .

$lim_{x \to \infty} \frac{4}{x (3 x^{2})}$

Step 5.2

Raise $x$ to the power of $1$ .

$lim_{x \to \infty} \frac{4}{3 (x^{1} x^{2})}$

Step 5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to \infty} \frac{4}{3 x^{1 + 2}}$

Step 5.4

Add $1$ and $2$ .

$lim_{x \to \infty} \frac{4}{3 x^{3}}$

Step 6

Move the term $\frac{4}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{4}{3} lim_{x \to \infty} \frac{1}{x^{3}}$

Step 7

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

$\frac{4}{3} \cdot 0$

Step 8

Multiply $\frac{4}{3}$ by $0$ .

$0$