Calculus Examples

$lim_{x \to \infty} \frac{x^{2} + 3 x + 12}{- 6 \ln (x^{3})}$

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.

$\frac{lim_{x \to \infty} x^{2} + 3 x + 12}{lim_{x \to \infty} - 6 \ln (x^{3})}$

Step 1.2

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

$\frac{\infty}{lim_{x \to \infty} - 6 \ln (x^{3})}$

Step 1.3

Since the function $\ln (x^{3})$ approaches $\infty$ , the negative constant $- 6$ times the function approaches $- \infty$ .

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Step 1.3.1

Consider the limit with the constant multiple $- 6$ removed.

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

Step 1.3.2

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

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

Step 1.3.3

Since the function $\ln (x^{3})$ approaches $\infty$ , the negative constant $- 6$ times the function approaches $- \infty$ .

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

Step 1.3.4

Infinity divided by infinity is undefined.

Undefined

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

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.

$lim_{x \to \infty} \frac{x^{2} + 3 x + 12}{- 6 \ln (x^{3})} = lim_{x \to \infty} \frac{\frac{d}{d x} [x^{2} + 3 x + 12]}{\frac{d}{d x} [- 6 \ln (x^{3})]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [x^{2} + 3 x + 12]}{\frac{d}{d x} [- 6 \ln (x^{3})]}$

Step 3.2

By the Sum Rule, the derivative of $x^{2} + 3 x + 12$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [12]$ .

$lim_{x \to \infty} \frac{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [12]}{\frac{d}{d x} [- 6 \ln (x^{3})]}$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to \infty} \frac{2 x + \frac{d}{d x} [3 x] + \frac{d}{d x} [12]}{\frac{d}{d x} [- 6 \ln (x^{3})]}$

Step 3.4

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 3.4.1

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

$lim_{x \to \infty} \frac{2 x + 3 \frac{d}{d x} [x] + \frac{d}{d x} [12]}{\frac{d}{d x} [- 6 \ln (x^{3})]}$

Step 3.4.2

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

$lim_{x \to \infty} \frac{2 x + 3 \cdot 1 + \frac{d}{d x} [12]}{\frac{d}{d x} [- 6 \ln (x^{3})]}$

Step 3.4.3

Multiply $3$ by $1$ .

$lim_{x \to \infty} \frac{2 x + 3 + \frac{d}{d x} [12]}{\frac{d}{d x} [- 6 \ln (x^{3})]}$

Step 3.5

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$lim_{x \to \infty} \frac{2 x + 3 + 0}{\frac{d}{d x} [- 6 \ln (x^{3})]}$

Step 3.6

Add $2 x + 3$ and $0$ .

$lim_{x \to \infty} \frac{2 x + 3}{\frac{d}{d x} [- 6 \ln (x^{3})]}$

Step 3.7

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 \ln (x^{3})$ with respect to $x$ is $- 6 \frac{d}{d x} [\ln (x^{3})]$ .

$lim_{x \to \infty} \frac{2 x + 3}{- 6 \frac{d}{d x} [\ln (x^{3})]}$

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

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Step 3.8.1

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

$lim_{x \to \infty} \frac{2 x + 3}{- 6 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{3}])}$

Step 3.8.2

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

$lim_{x \to \infty} \frac{2 x + 3}{- 6 (\frac{1}{u} \frac{d}{d x} [x^{3}])}$

Step 3.8.3

Replace all occurrences of $u$ with $x^{3}$ .

$lim_{x \to \infty} \frac{2 x + 3}{- 6 (\frac{1}{x^{3}} \frac{d}{d x} [x^{3}])}$

Step 3.9

Combine $\frac{1}{x^{3}}$ and $- 6$ .

$lim_{x \to \infty} \frac{2 x + 3}{\frac{- 6}{x^{3}} \frac{d}{d x} [x^{3}]}$

Step 3.10

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{2 x + 3}{- \frac{6}{x^{3}} \frac{d}{d x} [x^{3}]}$

Step 3.11

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

$lim_{x \to \infty} \frac{2 x + 3}{- \frac{6}{x^{3}} (3 x^{2})}$

Step 3.12

Multiply $3$ by $- 1$ .

$lim_{x \to \infty} \frac{2 x + 3}{- 3 \frac{6}{x^{3}} x^{2}}$

Step 3.13

Combine $- 3$ and $\frac{6}{x^{3}}$ .

$lim_{x \to \infty} \frac{2 x + 3}{\frac{- 3 \cdot 6}{x^{3}} x^{2}}$

Step 3.14

Multiply $- 3$ by $6$ .

$lim_{x \to \infty} \frac{2 x + 3}{\frac{- 18}{x^{3}} x^{2}}$

Step 3.15

Combine $\frac{- 18}{x^{3}}$ and $x^{2}$ .

$lim_{x \to \infty} \frac{2 x + 3}{\frac{- 18 x^{2}}{x^{3}}}$

Step 3.16

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 3.16.1

Factor $x^{2}$ out of $- 18 x^{2}$ .

$lim_{x \to \infty} \frac{2 x + 3}{\frac{x^{2} \cdot - 18}{x^{3}}}$

Step 3.16.2

Cancel the common factors.

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Step 3.16.2.1

Factor $x^{2}$ out of $x^{3}$ .

$lim_{x \to \infty} \frac{2 x + 3}{\frac{x^{2} \cdot - 18}{x^{2} x}}$

Step 3.16.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{2 x + 3}{\frac{x^{2} \cdot - 18}{x^{2} x}}$

Step 3.16.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{2 x + 3}{\frac{- 18}{x}}$

Step 3.17

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{2 x + 3}{- \frac{18}{x}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to \infty} (2 x + 3) (- \frac{x}{18})$

Step 5

Apply the distributive property.

$lim_{x \to \infty} 2 x (- \frac{x}{18}) + 3 (- \frac{x}{18})$

Step 6

Simplify the expression.

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Step 6.1

Move $x$ .

$lim_{x \to \infty} 2 \cdot - 1 x \frac{x}{18} + 3 (- \frac{x}{18})$

Step 6.2

Multiply $2$ by $- 1$ .

$lim_{x \to \infty} - 2 x \frac{x}{18} + 3 \cdot - 1 \frac{x}{18}$

Step 7

Combine $- 2 x$ and $\frac{x}{18}$ .

$lim_{x \to \infty} \frac{- 2 x \cdot x}{18} + 3 \cdot - 1 \frac{x}{18}$

Step 8

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

$lim_{x \to \infty} \frac{- 2 (x^{1} x)}{18} + 3 \cdot - 1 \frac{x}{18}$

Step 9

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

$lim_{x \to \infty} \frac{- 2 (x^{1} x^{1})}{18} + 3 \cdot - 1 \frac{x}{18}$

Step 10

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

$lim_{x \to \infty} \frac{- 2 x^{1 + 1}}{18} + 3 \cdot - 1 \frac{x}{18}$

Step 11

Combine fractions.

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Step 11.1

Add $1$ and $1$ .

$lim_{x \to \infty} \frac{- 2 x^{2}}{18} + 3 \cdot - 1 \frac{x}{18}$

Step 11.2

Multiply $3$ by $- 1$ .

$lim_{x \to \infty} \frac{- 2 x^{2}}{18} - 3 \frac{x}{18}$

Step 11.3

Combine $- 3$ and $\frac{x}{18}$ .

$lim_{x \to \infty} \frac{- 2 x^{2}}{18} + \frac{- 3 x}{18}$

Step 12

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

$\infty$