Find the Asymptotes (5x)/(1-x)
The problem is asking to determine the lines, if any, that the graph of the function (5x)/(1-x) approaches as the variable x tends toward positive or negative infinity, or as x approaches certain critical values where the function is not defined. In this case, the function is a rational function, and the quest involves finding vertical and possibly horizontal or oblique asymptotes, which are the lines that the graph gets infinitely close to, but does not intersect, at extreme values of x or at points of discontinuity.
$\frac{5 x}{1 - x}$
Answer
Solution:
Step:1 Identify the values of $x$ that cause the function $f(x) = \frac{5x}{1 - x}$ to be undefined. These values are the vertical asymptotes. For this function, the vertical asymptote is at $x = 1$.
Step:2 Examine the degrees of the polynomial in the numerator and the polynomial in the denominator of a rational function $f(x) = \frac{ax^n}{bx^m}$. The following rules apply for horizontal asymptotes:
If the degree of the numerator ($n$) is less than the degree of the denominator ($m$), the horizontal asymptote is the x-axis, or $y = 0$.
If the degrees are equal ($n = m$), the horizontal asymptote is the line $y = \frac{a}{b}$.
If the degree of the numerator ($n$) is greater than the degree of the denominator ($m$), there is no horizontal asymptote, but there may be an oblique asymptote.
Step:3 Determine the degrees $n$ and $m$ for the given function. In this case, $n = 1$ and $m = 1$.
Step:4 Since the degrees of the numerator and denominator are equal ($n = m$), calculate the horizontal asymptote using the coefficients $a$ and $b$. Here, $a = 5$ and $b = -1$. Thus, the horizontal asymptote is $y = \frac{a}{b} = -5$.
Step:5 An oblique asymptote is not present since 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 Asymptote: $x = 1$ Horizontal Asymptote: $y = -5$ No Oblique Asymptote
Knowledge Notes:
To find the asymptotes of a rational function, one must understand the behavior of the function as it approaches certain critical points or as $x$ approaches infinity. The types of asymptotes to consider are:
Vertical Asymptotes: These occur at values of $x$ where the function becomes undefined, typically where the denominator is zero. To find vertical asymptotes, set the denominator equal to zero and solve for $x$.
Horizontal Asymptotes: These describe the behavior of the function as $x$ approaches infinity. The rules for finding horizontal asymptotes depend on the degrees of the numerator ($n$) and the denominator ($m$) of the rational function $f(x) = \frac{ax^n}{bx^m}$:
If $n < m$, the horizontal asymptote is $y = 0$.
If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively.
If $n > m$, there is no horizontal asymptote.
Oblique (Slant) Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. To find an oblique asymptote, perform polynomial long division of the numerator by the denominator. The quotient (ignoring the remainder) will give the equation of the oblique asymptote.
In the given problem, the function $f(x) = \frac{5x}{1 - x}$ has a vertical asymptote at $x = 1$ and a horizontal asymptote at $y = -5$. There is no oblique asymptote because the degrees of the numerator and denominator are equal.
