Find the Asymptotes (x-7)/(49x-x^3)
The question asks for the identification of the asymptotes of the function (x-7)/(49x-x^3). Asymptotes are lines which the graph of a function approaches as the inputs (or x-values) either increase indefinitely, decrease indefinitely, or approach a certain finite value. There are typically two kinds of asymptotes considered for rational functions: vertical and horizontal (or slant/oblique) asymptotes.
Vertical asymptotes occur where the function is undefined, typically at values of x which cause the denominator to be zero, provided the numerator does not also become zero at those points (thus potentially indicating a removable discontinuity instead).
Horizontal or oblique asymptotes are lines that the graph approaches as x goes to positive or negative infinity. These are found through different methods, such as polynomial long division or limits, depending on the degrees of the polynomials in the numerator and the denominator. The question does not request to include the calculations or specific methods, but only to identify which lines, if any, are asymptotes to the given function.
$\frac{x - 7}{49 x - x^{3}}$
Answer
Solution:
Step 1:
Identify the values of $x$ for which the function $\frac{x - 7}{49x - x^3}$ does not exist. These are the points where the denominator equals zero: $x = -7, x = 0, x = 7$.
Step 2:
Examine the behavior of $\frac{x - 7}{49x - x^3}$ as $x$ approaches $-7$. It tends towards $-\infty$ when approaching from the left and towards $\infty$ when approaching from the right. Thus, $x = -7$ is a vertical asymptote.
Step 3:
Observe the behavior of $\frac{x - 7}{49x - x^3}$ as $x$ approaches $0$. It heads towards $\infty$ when approaching from the left and towards $-\infty$ when approaching from the right. Hence, $x = 0$ is a vertical asymptote.
Step 4:
Compile a list of all vertical asymptotes found: $x = -7, 0$.
Step 5:
Understand the conditions for horizontal asymptotes in a rational function $R(x) = \frac{ax^n}{bx^m}$:
If $n < m$, the horizontal asymptote is the x-axis, $y = 0$.
If $n = m$, the horizontal asymptote is the line $y = \frac{a}{b}$.
If $n > m$, there are no horizontal asymptotes; instead, there may be an oblique asymptote.
Step 6:
Determine the degrees $n$ and $m$ of the numerator and denominator, respectively. Here, $n = 1$ and $m = 3$.
Step 7:
Since $n < m$, the horizontal asymptote is the x-axis, given by $y = 0$.
Step 8:
Conclude that there are no oblique asymptotes because the degree of the numerator is less than the degree of the denominator.
Step 9:
Summarize the asymptotes of the function:
Vertical Asymptotes: $x = -7, 0$ Horizontal Asymptote: $y = 0$ No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a rational function, one must first understand the different types of asymptotes:
Vertical Asymptotes: These occur at values of $x$ where the denominator of the rational function is zero, and the numerator is not zero. The function will approach infinity or negative infinity as it gets close to these points from either side.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator and the denominator ($n$ and $m$ respectively). If the degree of the numerator is less than the degree of the denominator, the x-axis ($y=0$) is the horizontal asymptote. If the degrees are equal, the horizontal asymptote is the line $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator.
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 by long division of the numerator by the denominator.
In the given problem, the rational function is $\frac{x - 7}{49x - x^3}$. To find the vertical asymptotes, we set the denominator equal to zero and solve for $x$. To determine the horizontal asymptote, we compare the degrees of the numerator and the denominator. Since the degree of the numerator ($n=1$) is less than the degree of the denominator ($m=3$), the horizontal asymptote is the x-axis, $y=0$. There are no oblique asymptotes because the degree of the numerator is not one more than the degree of the denominator.
