Find the Asymptotes (x-3)/(x^2+x-12)
The question is asking for the identification of any asymptotes associated with the function (x-3)/(x^2+x-12). An asymptote refers to a line that a graph of a function approaches but never actually reaches. For rational functions like the one given, asymptotes can be either vertical or horizontal (or, more rarely, oblique).
Specifically, a vertical asymptote occurs at values of x where the denominator of a function is equal to zero (provided the numerator isn't also zero at those points, as that might indicate a removable discontinuity instead). So for this function, you'll need to determine where the denominator, x^2+x-12, equals zero.
On the other hand, a horizontal asymptote relates to the end behavior of the function—how it behaves as x approaches infinity or negative infinity. This usually depends on the degrees of the polynomials in the numerator and the denominator. You would compare the highest power of x in the numerator and the denominator to determine if a horizontal asymptote exists and where it is.
Lastly, oblique asymptotes may occur when the degree of the numerator is exactly one greater than the degree of the denominator. However, that is not the case with this function since the degrees are the same. Hence, you won't have to look for oblique asymptotes in this instance.
$\frac{x - 3}{x^{2} + x - 12}$
Answer
Solution:
Step 1:
Identify the values of $x$ for which $\frac{x - 3}{x^{2} + x - 12}$ does not exist. These are $x = -4$ and $x = 3$.
Step 2:
Observe the behavior of $\frac{x - 3}{x^{2} + x - 12}$ as $x$ approaches $-4$. It tends towards $-\infty$ when approaching from the left and towards $\infty$ when approaching from the right, indicating a vertical asymptote at $x = -4$.
Step 3:
Examine the general form of a rational function $R(x) = \frac{a x^{n}}{b x^{m}}$ to determine horizontal asymptotes based on the degrees $n$ and $m$ of the numerator and denominator, respectively:
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, but potentially an oblique asymptote.
Step 4:
Calculate the degrees $n$ and $m$ for the numerator and denominator. Here, $n = 1$ and $m = 2$.
Step 5:
Since $n < m$, conclude that the horizontal asymptote is the x-axis, which is $y = 0$.
Step 6:
Determine that there is no oblique asymptote because the degree of the numerator is less than the degree of the denominator.
Step 7:
Compile the list of asymptotes for the function:
- Vertical Asymptotes: $x = -4$
- Horizontal Asymptotes: $y = 0$
- No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a rational function, we follow these steps:
Vertical Asymptotes: These occur at values of $x$ that make the denominator zero (as long as those values do not also make the numerator zero). To find them, set the denominator equal to zero and solve for $x$.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator and denominator ($n$ and $m$ respectively):
If the degree of the numerator ($n$) is less than the degree of the denominator ($m$), the horizontal asymptote is $y = 0$.
If the degrees are equal, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator.
If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
Oblique Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. In such cases, we perform long division to find the oblique asymptote.
Behavior Near Vertical Asymptotes: To determine the behavior of the function near a vertical asymptote, we examine the limits of the function as $x$ approaches the asymptote from the left and right.
Nonexistence of Asymptotes: If the degree of the numerator is more than one greater than the degree of the denominator, there are no horizontal or oblique asymptotes.
