Find the Asymptotes 5-3/x
The problem provided seems to involve finding the asymptotes of a rational function. Specifically, the function in question is \( f(x) = 5 - \frac{3}{x} \). The question is asking to determine the lines that the graph of this function approaches infinitely close to as the independent variable \( x \) either grows without bound or moves towards a specific value where the function is undefined.
Asymptotes can be vertical, horizontal, or oblique (slant), and they represent the behavior of the graph at the extreme ends of the x-values or near points where the function is not defined. To solve this problem, one would typically look for values of \( x \) which cause the denominator to be zero (for vertical asymptotes) and analyze the behavior of the function as \( x \) approaches infinity or negative infinity (for horizontal asymptotes). If a function has a degree of one more in the numerator than in the denominator, it might also have an oblique asymptote, which would be evaluated differently.
$5 - \frac{3}{x}$
Answer
Solution:
Step 1:
Identify the points at which the function $5 - \frac{3}{x}$ does not exist. This occurs when $x = 0$.
Step 2:
Examine the general form of a rational function $R(x) = \frac{a x^n}{b x^m}$, where $n$ is the highest power in the numerator and $m$ is the highest power in the denominator. The rules for horizontal asymptotes are:
If $n < m$, the horizontal asymptote is $y = 0$.
If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$.
If $n > m$, there is no horizontal asymptote; instead, look for an oblique asymptote.
Step 3:
Determine the values of $n$ and $m$. For our function, $n = 0$ and $m = 1$.
Step 4:
Since $n = m$, we use the rule that the horizontal asymptote is $y = \frac{a}{b}$. Here, $a = 5$ and $b = 1$, so $y = \frac{5}{1} = 5$.
Step 5:
An oblique asymptote is not present because the degree of the numerator is not greater than the degree of the denominator.
Step 6:
Compile the list of asymptotes for the function:
- Vertical Asymptotes: $x = 0$
- Horizontal Asymptotes: $y = 5$
- No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a function, one must understand the behavior of the function as it approaches certain critical points or infinity. Asymptotes are lines that the graph of a function approaches but never touches.
Vertical Asymptotes: These occur at values of $x$ where the function becomes undefined, typically where the denominator of a rational function is zero.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator and the denominator ($n$ and $m$ respectively) in a rational function. If the degree of the numerator is less than the degree of the denominator, the x-axis is the horizontal asymptote. If the degrees are equal, the horizontal asymptote is found by dividing the leading coefficients. If the numerator's degree is greater, there is no horizontal asymptote.
Oblique Asymptotes: If the degree of the numerator is exactly one more than the degree of the denominator, the function may have an oblique (or slant) asymptote, which is found using long division or synthetic division.
Rational Functions: These are functions expressed as the ratio of two polynomials. The general form is $R(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials.
Undefined Points: A function is undefined at points where it would require division by zero or result in an indeterminate form.
In the given problem, the function $5 - \frac{3}{x}$ is a rational function with a constant numerator and a linear denominator. The process involves identifying where the function is undefined, applying rules for horizontal asymptotes based on the degrees of the numerator and denominator, and concluding that there are no oblique asymptotes due to the degrees of the polynomials involved.
