Trigonometry Examples

$f (x) = - 2 {(x - 4)}^{2 (x^{2 - 25})}$

Step 1

Find where the expression $- 2 {(x - 4)}^{\frac{2}{x^{23}}}$ is undefined.

$x = 0$

Step 2

Evaluate $lim_{x \to \infty} - 2 {(x - 4)}^{\frac{2}{x^{23}}}$ to find the horizontal asymptote.

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Step 2.1

Move the term $- 2$ outside of the limit because it is constant with respect to $x$ .

$- 2 lim_{x \to \infty} {(x - 4)}^{\frac{2}{x^{23}}}$

Step 2.2

Use the properties of logarithms to simplify the limit.

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Step 2.2.1

Rewrite ${(x - 4)}^{\frac{2}{x^{23}}}$ as $e^{\ln ({(x - 4)}^{\frac{2}{x^{23}}})}$ .

$- 2 lim_{x \to \infty} e^{\ln ({(x - 4)}^{\frac{2}{x^{23}}})}$

Step 2.2.2

Expand $\ln ({(x - 4)}^{\frac{2}{x^{23}}})$ by moving $\frac{2}{x^{23}}$ outside the logarithm.

$- 2 lim_{x \to \infty} e^{\frac{2}{x^{23}} \ln (x - 4)}$

Step 2.3

Evaluate the limit.

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Step 2.3.1

Move the limit into the exponent.

$- 2 e^{lim_{x \to \infty} \frac{2}{x^{23}} \ln (x - 4)}$

Step 2.3.2

Combine $\frac{2}{x^{23}}$ and $\ln (x - 4)$ .

$- 2 e^{lim_{x \to \infty} \frac{2 \ln (x - 4)}{x^{23}}}$

Step 2.3.3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$- 2 e^{2 lim_{x \to \infty} \frac{\ln (x - 4)}{x^{23}}}$

Step 2.4

Apply L'Hospital's rule.

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Step 2.4.1

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

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Step 2.4.1.1

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

$- 2 e^{2 \frac{lim_{x \to \infty} \ln (x - 4)}{lim_{x \to \infty} x^{23}}}$

Step 2.4.1.2

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

$- 2 e^{2 \frac{\infty}{lim_{x \to \infty} x^{23}}}$

Step 2.4.1.3

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

$- 2 e^{2 \frac{\infty}{\infty}}$

Step 2.4.1.4

Infinity divided by infinity is undefined.

Undefined

$- 2 e^{2 \frac{\infty}{\infty}}$

Step 2.4.2

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.

$lim_{x \to \infty} \frac{\ln (x - 4)}{x^{23}} = lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (x - 4)]}{\frac{d}{d x} [x^{23}]}$

Step 2.4.3

Find the derivative of the numerator and denominator.

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Step 2.4.3.1

Differentiate the numerator and denominator.

$- 2 e^{2 lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (x - 4)]}{\frac{d}{d x} [x^{23}]}}$

Step 2.4.3.2

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 2.4.3.2.1

To apply the Chain Rule, set $u$ as $x - 4$ .

$- 2 e^{2 lim_{x \to \infty} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x - 4]}{\frac{d}{d x} [x^{23}]}}$

Step 2.4.3.2.2

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

$- 2 e^{2 lim_{x \to \infty} \frac{\frac{1}{u} \frac{d}{d x} [x - 4]}{\frac{d}{d x} [x^{23}]}}$

Step 2.4.3.2.3

Replace all occurrences of $u$ with $x - 4$ .

$- 2 e^{2 lim_{x \to \infty} \frac{\frac{1}{x - 4} \frac{d}{d x} [x - 4]}{\frac{d}{d x} [x^{23}]}}$

Step 2.4.3.3

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$- 2 e^{2 lim_{x \to \infty} \frac{\frac{1}{x - 4} (\frac{d}{d x} [x] + \frac{d}{d x} [- 4])}{\frac{d}{d x} [x^{23}]}}$

Step 2.4.3.4

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

$- 2 e^{2 lim_{x \to \infty} \frac{\frac{1}{x - 4} (1 + \frac{d}{d x} [- 4])}{\frac{d}{d x} [x^{23}]}}$

Step 2.4.3.5

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$- 2 e^{2 lim_{x \to \infty} \frac{\frac{1}{x - 4} (1 + 0)}{\frac{d}{d x} [x^{23}]}}$

Step 2.4.3.6

Add $1$ and $0$ .

$- 2 e^{2 lim_{x \to \infty} \frac{\frac{1}{x - 4} \cdot 1}{\frac{d}{d x} [x^{23}]}}$

Step 2.4.3.7

Multiply $\frac{1}{x - 4}$ by $1$ .

$- 2 e^{2 lim_{x \to \infty} \frac{\frac{1}{x - 4}}{\frac{d}{d x} [x^{23}]}}$

Step 2.4.3.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 23$ .

$- 2 e^{2 lim_{x \to \infty} \frac{\frac{1}{x - 4}}{23 x^{22}}}$

Step 2.4.4

Multiply the numerator by the reciprocal of the denominator.

$- 2 e^{2 lim_{x \to \infty} \frac{1}{x - 4} \cdot \frac{1}{23 x^{22}}}$

Step 2.4.5

Multiply $\frac{1}{x - 4}$ by $\frac{1}{23 x^{22}}$ .

$- 2 e^{2 lim_{x \to \infty} \frac{1}{(x - 4) \cdot 23 x^{22}}}$

Step 2.5

Move the term $\frac{1}{23}$ outside of the limit because it is constant with respect to $x$ .

$- 2 e^{2 (\frac{1}{23}) lim_{x \to \infty} \frac{1}{(x - 4) x^{22}}}$

Step 2.6

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{(x - 4) x^{22}}$ approaches $0$ .

$- 2 e^{2 (\frac{1}{23}) \cdot 0}$

Step 2.7

Simplify the answer.

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Step 2.7.1

Combine $2$ and $\frac{1}{23}$ .

$- 2 e^{\frac{2}{23} \cdot 0}$

Step 2.7.2

Multiply $\frac{2}{23}$ by $0$ .

$- 2 e^{0}$

Step 2.7.3

Anything raised to $0$ is $1$ .

$- 2 \cdot 1$

Step 2.7.4

Multiply $- 2$ by $1$ .

$- 2$

Step 3

List the horizontal asymptotes:

$y = - 2$

Step 4

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

Step 5

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = - 2$

No Oblique Asymptotes

Step 6