Precalculus Examples

$f (x) = \frac{1}{10^{x}} + 4$

Step 1

Find where the expression $\frac{1}{10^{x}} + 4$ 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 $lim_{x \to \infty} \frac{1 + 4 \cdot 10^{x}}{10^{x}}$ 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.

$\frac{lim_{x \to \infty} 1 + 4 \cdot 10^{x}}{lim_{x \to \infty} 10^{x}}$

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 $\infty$ .

$\frac{lim_{x \to \infty} 1 + lim_{x \to \infty} 4 \cdot 10^{x}}{lim_{x \to \infty} 10^{x}}$

Step 3.1.1.2.1.2

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

$\frac{1 + lim_{x \to \infty} 4 \cdot 10^{x}}{lim_{x \to \infty} 10^{x}}$

Step 3.1.1.2.2

Since the function $10^{x}$ approaches $\infty$ , the positive constant $4$ times the function also approaches $\infty$ .

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Step 3.1.1.2.2.1

Consider the limit with the constant multiple $4$ removed.

$\frac{1 + lim_{x \to \infty} 10^{x}}{lim_{x \to \infty} 10^{x}}$

Step 3.1.1.2.2.2

Since the exponent $x$ approaches $\infty$ , the quantity $10^{x}$ approaches $\infty$ .

$\frac{1 + \infty}{lim_{x \to \infty} 10^{x}}$

Step 3.1.1.2.3

Infinity plus or minus a number is infinity.

$\frac{\infty}{lim_{x \to \infty} 10^{x}}$

Step 3.1.1.3

Since the exponent $x$ approaches $\infty$ , the quantity $10^{x}$ 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.

$lim_{x \to \infty} \frac{1 + 4 \cdot 10^{x}}{10^{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [1 + 4 \cdot 10^{x}]}{\frac{d}{d x} [10^{x}]}$

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.

$lim_{x \to \infty} \frac{\frac{d}{d x} [1 + 4 \cdot 10^{x}]}{\frac{d}{d x} [10^{x}]}$

Step 3.1.3.2

By the Sum Rule, the derivative of $1 + 4 \cdot 10^{x}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [4 \cdot 10^{x}]$ .

$lim_{x \to \infty} \frac{\frac{d}{d x} [1] + \frac{d}{d x} [4 \cdot 10^{x}]}{\frac{d}{d x} [10^{x}]}$

Step 3.1.3.3

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

$lim_{x \to \infty} \frac{0 + \frac{d}{d x} [4 \cdot 10^{x}]}{\frac{d}{d x} [10^{x}]}$

Step 3.1.3.4

Evaluate $\frac{d}{d x} [4 \cdot 10^{x}]$ .

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Step 3.1.3.4.1

Since $4$ is constant with respect to $x$ , the derivative of $4 \cdot 10^{x}$ with respect to $x$ is $4 \frac{d}{d x} [10^{x}]$ .

$lim_{x \to \infty} \frac{0 + 4 \frac{d}{d x} [10^{x}]}{\frac{d}{d x} [10^{x}]}$

Step 3.1.3.4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $10$ .

$lim_{x \to \infty} \frac{0 + 4 \cdot 10^{x} \ln (10)}{\frac{d}{d x} [10^{x}]}$

Step 3.1.3.5

Add $0$ and $4 \cdot 10^{x} \ln (10)$ .

$lim_{x \to \infty} \frac{4 \cdot 10^{x} \ln (10)}{\frac{d}{d x} [10^{x}]}$

Step 3.1.3.6

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $10$ .

$lim_{x \to \infty} \frac{4 \cdot 10^{x} \ln (10)}{10^{x} \ln (10)}$

Step 3.1.4

Reduce.

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Step 3.1.4.1

Cancel the common factor of $10^{x}$ .

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Step 3.1.4.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{4 \cdot 10^{x} \ln (10)}{10^{x} \ln (10)}$

Step 3.1.4.1.2

Rewrite the expression.

$lim_{x \to \infty} \frac{4 \ln (10)}{\ln (10)}$

Step 3.1.4.2

Cancel the common factor of $\ln (10)$ .

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Step 3.1.4.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{4 \ln (10)}{\ln (10)}$

Step 3.1.4.2.2

Divide $4$ by $1$ .

$lim_{x \to \infty} 4$

Step 3.2

Evaluate the limit of $4$ which is constant as $x$ approaches $\infty$ .

$4$

Step 4

List the horizontal asymptotes:

$y = 4$

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: $y = 4$

No Oblique Asymptotes

Step 7

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