Calculus Examples

$\frac{1 - 2 \ln (x)}{x^{3}}$

Step 1

Find the asymptotes.

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Step 1.1

Find where the expression $\frac{1 - \ln (x^{2})}{x^{3}}$ is undefined.

$x = 0$

Step 1.2

Evaluate $lim_{x \to \infty} \frac{1 - \ln (x^{2})}{x^{3}}$ to find the horizontal asymptote.

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Step 1.2.1

Apply L'Hospital's rule.

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Step 1.2.1.1

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

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Step 1.2.1.1.1

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

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

Step 1.2.1.1.2

Evaluate the limit of the numerator.

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Step 1.2.1.1.2.1

Evaluate the limit.

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Step 1.2.1.1.2.1.1

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

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

Step 1.2.1.1.2.1.2

Evaluate the limit of $1$ which is constant as $x$ approaches $\infty$ .

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

Step 1.2.1.1.2.2

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

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

Step 1.2.1.1.2.3

Simplify the answer.

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Step 1.2.1.1.2.3.1

A non-zero constant times infinity is infinity.

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

Step 1.2.1.1.2.3.2

Infinity plus or minus a number is infinity.

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

Step 1.2.1.1.3

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

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

Step 1.2.1.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 1.2.1.2

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.

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

Step 1.2.1.3

Find the derivative of the numerator and denominator.

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Step 1.2.1.3.1

Differentiate the numerator and denominator.

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

Step 1.2.1.3.2

By the Sum Rule, the derivative of $1 - \ln (x^{2})$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \ln (x^{2})]$ .

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

Step 1.2.1.3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.2.1.3.4

Evaluate $\frac{d}{d x} [- \ln (x^{2})]$ .

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Step 1.2.1.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \ln (x^{2})$ with respect to $x$ is $- \frac{d}{d x} [\ln (x^{2})]$ .

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

Step 1.2.1.3.4.2

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 $g (x) = x^{2}$ .

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Step 1.2.1.3.4.2.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

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

Step 1.2.1.3.4.2.2

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

$lim_{x \to \infty} \frac{0 - (\frac{1}{u} \frac{d}{d x} [x^{2}])}{\frac{d}{d x} [x^{3}]}$

Step 1.2.1.3.4.2.3

Replace all occurrences of $u$ with $x^{2}$ .

$lim_{x \to \infty} \frac{0 - (\frac{1}{x^{2}} \frac{d}{d x} [x^{2}])}{\frac{d}{d x} [x^{3}]}$

Step 1.2.1.3.4.3

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

$lim_{x \to \infty} \frac{0 - (\frac{1}{x^{2}} (2 x))}{\frac{d}{d x} [x^{3}]}$

Step 1.2.1.3.4.4

Combine $2$ and $\frac{1}{x^{2}}$ .

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

Step 1.2.1.3.4.5

Combine $\frac{2}{x^{2}}$ and $x$ .

$lim_{x \to \infty} \frac{0 - \frac{2 x}{x^{2}}}{\frac{d}{d x} [x^{3}]}$

Step 1.2.1.3.4.6

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

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Step 1.2.1.3.4.6.1

Factor $x$ out of $2 x$ .

$lim_{x \to \infty} \frac{0 - \frac{x \cdot 2}{x^{2}}}{\frac{d}{d x} [x^{3}]}$

Step 1.2.1.3.4.6.2

Cancel the common factors.

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Step 1.2.1.3.4.6.2.1

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

$lim_{x \to \infty} \frac{0 - \frac{x \cdot 2}{x \cdot x}}{\frac{d}{d x} [x^{3}]}$

Step 1.2.1.3.4.6.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{0 - \frac{x \cdot 2}{x \cdot x}}{\frac{d}{d x} [x^{3}]}$

Step 1.2.1.3.4.6.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{0 - \frac{2}{x}}{\frac{d}{d x} [x^{3}]}$

Step 1.2.1.3.5

Subtract $\frac{2}{x}$ from $0$ .

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

Step 1.2.1.3.6

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

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

Step 1.2.1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.1.5

Combine factors.

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Step 1.2.1.5.1

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

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

Step 1.2.1.5.2

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

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

Step 1.2.1.5.3

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

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

Step 1.2.1.5.4

Add $1$ and $2$ .

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

Step 1.2.2

Evaluate the limit.

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Step 1.2.2.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.2.2.2

Move the term $\frac{2}{3}$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.2.3

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

$- \frac{2}{3} \cdot 0$

Step 1.2.4

Multiply $- \frac{2}{3} \cdot 0$ .

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Step 1.2.4.1

Multiply $0$ by $- 1$ .

$0 (\frac{2}{3})$

Step 1.2.4.2

Multiply $0$ by $\frac{2}{3}$ .

$0$

Step 1.3

List the horizontal asymptotes:

$y = 0$

Step 1.4

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

Step 1.5

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 0$

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 0$

Step 2

Find the point at $x = 1$ .

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Step 2.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{1 - 2 \ln (1)}{{(1)}^{3}}$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

The natural logarithm of $1$ is $0$ .

$f (1) = \frac{1 - 2 \cdot 0}{1^{3}}$

Step 2.2.1.2

Multiply $- 2$ by $0$ .

$f (1) = \frac{1 + 0}{1^{3}}$

Step 2.2.1.3

Add $1$ and $0$ .

$f (1) = \frac{1}{1^{3}}$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

One to any power is one.

$f (1) = \frac{1}{1}$

Step 2.2.2.2

Divide $1$ by $1$ .

$f (1) = 1$

Step 2.2.3

The final answer is $1$ .

$1$

Step 2.3

Convert $1$ to decimal.

$y = 1$

Step 3

Find the point at $x = 2$ .

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Step 3.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = \frac{1 - 2 \ln (2)}{{(2)}^{3}}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify the numerator.

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Step 3.2.1.1

Simplify $- 2 \ln (2)$ by moving $2$ inside the logarithm.

$f (2) = \frac{1 - \ln (2^{2})}{2^{3}}$

Step 3.2.1.2

Raise $2$ to the power of $2$ .

$f (2) = \frac{1 - \ln (4)}{2^{3}}$

Step 3.2.2

Raise $2$ to the power of $3$ .

$f (2) = \frac{1 - \ln (4)}{8}$

Step 3.2.3

The final answer is $\frac{1 - \ln (4)}{8}$ .

$\frac{1 - \ln (4)}{8}$

Step 3.3

Convert $\frac{1 - \ln (4)}{8}$ to decimal.

$y = - 0.04828679$

Step 4

Find the point at $x = 3$ .

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Step 4.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{1 - 2 \ln (3)}{{(3)}^{3}}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify the numerator.

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Step 4.2.1.1

Simplify $- 2 \ln (3)$ by moving $2$ inside the logarithm.

$f (3) = \frac{1 - \ln (3^{2})}{3^{3}}$

Step 4.2.1.2

Raise $3$ to the power of $2$ .

$f (3) = \frac{1 - \ln (9)}{3^{3}}$

Step 4.2.2

Raise $3$ to the power of $3$ .

$f (3) = \frac{1 - \ln (9)}{27}$

Step 4.2.3

The final answer is $\frac{1 - \ln (9)}{27}$ .

$\frac{1 - \ln (9)}{27}$

Step 4.3

Convert $\frac{1 - \ln (9)}{27}$ to decimal.

$y = - 0.04434165$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 1), (2, - 0.04828679), (3, - 0.04434165)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 1 \\ 2 & - 0.048 \\ 3 & - 0.044 \end{array}$

Step 6