Find the Asymptotes (2x^2+6x-5)/(7x^2+3)
The question is asking for the identification of the asymptotes of the function given by the formula (2x^2 + 6x - 5) / (7x^2 + 3). Asymptotes are lines that the graph of the function approaches but never actually touches as x approaches positive or negative infinity. There are different types of asymptotes, such as vertical, horizontal, or oblique (slant) asymptotes. The problem requires analyzing the function to determine which of these types of asymptotes exist for this particular rational function based on the degrees of the polynomials in the numerator and the denominator and their leading coefficients.
$\frac{\left(\right. 2 x^{2} + 6 x - 5 \left.\right)}{7 x^{2} + 3}$
Answer
Solution:
Step 1:
Identify values that cause the function $\frac{2x^2 + 6x - 5}{7x^2 + 3}$ to be undefined. The function is defined for all real numbers since the denominator never equals zero.
Step 2:
Determine the vertical asymptotes, which are found where the function is not continuous. In this case, there are no vertical asymptotes.
Step 3:
Analyze the degrees of the numerator and denominator in the function $R(x) = \frac{ax^n}{bx^m}$. The rules for horizontal asymptotes are as follows:
If $n < m$, the horizontal asymptote is $y = 0$.
If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$.
If $n > m$, there are no horizontal asymptotes, but possibly an oblique asymptote.
Step 4:
Calculate the degrees of the numerator and denominator. For our function, $n = 2$ and $m = 2$.
Step 5:
Since the degrees are equal ($n = m$), the horizontal asymptote is found using $y = \frac{a}{b}$. For our function, $a = 2$ and $b = 7$, so $y = \frac{2}{7}$.
Step 6:
Check for the presence of an oblique asymptote. Since the degree of the numerator is not greater than the degree of the denominator, there is no oblique asymptote.
Step 7:
Compile the list of asymptotes for the function:
- No Vertical Asymptotes
- Horizontal Asymptote: $y = \frac{2}{7}$
- No Oblique Asymptotes
Knowledge Notes:
The concept of asymptotes is related to the behavior of graphs of functions as they approach certain lines or infinity. Asymptotes can be vertical, horizontal, or oblique.
Vertical Asymptotes: These occur at values of $x$ where the function approaches infinity. They are typically found where the denominator of a rational function equals zero, but the numerator does not.
Horizontal Asymptotes: These are horizontal lines that the graph of the function approaches as $x$ goes to infinity or negative infinity. The rules for finding horizontal asymptotes depend on the degrees of the numerator ($n$) and denominator ($m$) of the rational function:
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 Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. The oblique asymptote can be found using polynomial long division to divide the numerator by the denominator.
In the given problem, we are dealing with a rational function where the degrees of the numerator and denominator are equal, leading to a horizontal asymptote at $y = \frac{a}{b}$. There are no vertical or oblique asymptotes because the denominator does not have real roots, and the degree of the numerator is not greater than that of the denominator.
