Trigonometry Examples

$g (x) = \frac{1}{2^{x - 3}} + 1$

Step 1

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

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

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

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Step 3.1

Apply L'Hospital's rule.

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Step 3.1.1

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

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Step 3.1.1.1

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

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

Step 3.1.1.2

Evaluate the limit of the numerator.

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Step 3.1.1.2.1

Evaluate the limit.

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Step 3.1.1.2.1.1

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

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

Step 3.1.1.2.1.2

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

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

Step 3.1.1.2.2

Since the exponent $x - 3$ approaches $\infty$ , the quantity $2^{x - 3}$ approaches $\infty$ .

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

Step 3.1.1.2.3

Infinity plus or minus a number is infinity.

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

Step 3.1.1.3

Since the exponent $x - 3$ approaches $\infty$ , the quantity $2^{x - 3}$ approaches $\infty$ .

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

Step 3.1.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 3.1.2

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.

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

Step 3.1.3

Find the derivative of the numerator and denominator.

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Step 3.1.3.1

Differentiate the numerator and denominator.

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

Step 3.1.3.2

By the Sum Rule, the derivative of $1 + 2^{x - 3}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [2^{x - 3}]$ .

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

Step 3.1.3.3

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

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

Step 3.1.3.4

Evaluate $\frac{d}{d x} [2^{x - 3}]$ .

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Step 3.1.3.4.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 2^{x}$ and $g (x) = x - 3$ .

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Step 3.1.3.4.1.1

To apply the Chain Rule, set $u$ as $x - 3$ .

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

Step 3.1.3.4.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $2$ .

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

Step 3.1.3.4.1.3

Replace all occurrences of $u$ with $x - 3$ .

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

Step 3.1.3.4.2

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

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

Step 3.1.3.4.3

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

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

Step 3.1.3.4.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 3.1.3.4.5

Add $1$ and $0$ .

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

Step 3.1.3.4.6

Multiply $2^{x - 3}$ by $1$ .

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

Step 3.1.3.5

Add $0$ and $2^{x - 3} \ln (2)$ .

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

Step 3.1.3.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 2^{x}$ and $g (x) = x - 3$ .

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Step 3.1.3.6.1

To apply the Chain Rule, set $u$ as $x - 3$ .

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

Step 3.1.3.6.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $2$ .

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

Step 3.1.3.6.3

Replace all occurrences of $u$ with $x - 3$ .

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

Step 3.1.3.7

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

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

Step 3.1.3.8

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

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

Step 3.1.3.9

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$lim_{x \to \infty} \frac{2^{x - 3} \ln (2)}{2^{x - 3} \ln (2) (1 + 0)}$

Step 3.1.3.10

Add $1$ and $0$ .

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

Step 3.1.3.11

Multiply $2^{x - 3}$ by $1$ .

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

Step 3.1.4

Reduce.

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Step 3.1.4.1

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

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Step 3.1.4.1.1

Cancel the common factor.

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

Step 3.1.4.1.2

Rewrite the expression.

$lim_{x \to \infty} \frac{\ln (2)}{\ln (2)}$

Step 3.1.4.2

Cancel the common factor of $\ln (2)$ .

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Step 3.1.4.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\ln (2)}{\ln (2)}$

Step 3.1.4.2.2

Rewrite the expression.

$lim_{x \to \infty} 1$

Step 3.2

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

$1$

Step 4

List the horizontal asymptotes:

$y = 1$

Step 5

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 6

This is the set of all asymptotes.

No Vertical Asymptotes

Horizontal Asymptotes: $y = 1$

No Oblique Asymptotes

Step 7