Find the Asymptotes (5x^2)/(x^2+2)
The question asks for the identification of the asymptotes of the function (5x^2)/(x^2+2). This typically involves looking for lines that the graph of the function approaches but never actually reaches. The asymptotes can be vertical, where the function goes to infinity as x approaches a certain value, or horizontal, where the y-value approaches a constant value as x goes to infinity or negative infinity. Horizontal asymptotes in rational functions like this one are found by comparing the degrees of the polynomials in the numerator and the denominator. Vertical asymptotes often occur where the denominator is zero, as long as it doesn't also make the numerator zero at the same point. The task is to calculate or deduce these lines from the given function.
$\frac{5 x^{2}}{x^{2} + 2}$
Answer
Solution:
Step:1 Determine the values for which the function $\frac{5x^2}{x^2+2}$ does not exist. The function is defined for all real numbers except for those that make the denominator zero. In this instance, the denominator never equals zero.
Step:2 Identify any vertical asymptotes by locating points where the function approaches infinity. There are no vertical asymptotes for this function.
Step:3 Analyze the degrees of the polynomial in the numerator and the denominator of the function $R(x) = \frac{ax^n}{bx^m}$, where $n$ is the degree of the numerator and $m$ is the degree of the denominator. 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 potentially an oblique asymptote).
Step:4 Calculate the values of $n$ and $m$.
$n = 2$ $m = 2$
Step:5 Since $n$ equals $m$, the horizontal asymptote is found using $y = \frac{a}{b}$, with $a = 5$ and $b = 1$.
$y = 5$
Step:6 Determine the presence of any oblique asymptotes. Since the degree of the numerator is not greater than the degree of the denominator, there are no oblique asymptotes.
Step:7 Compile the list of asymptotes for the function.
No Vertical Asymptotes Horizontal Asymptote: $y = 5$ No Oblique Asymptotes
Knowledge Notes:
Asymptotes are lines that a graph approaches but never actually touches. They can be vertical, horizontal, or oblique (slant).
Vertical Asymptotes occur where the function goes to infinity as the input approaches a certain value. This typically happens when the denominator of a rational function is zero.
Horizontal Asymptotes are found by comparing the degrees of the numerator and the denominator in a rational function. If the degree of the numerator ($n$) is less than the degree of the denominator ($m$), the horizontal asymptote is $y=0$. If $n$ equals $m$, the horizontal asymptote is $y=\frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively. If $n$ is greater than $m$, there is no horizontal asymptote.
Oblique Asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. In such cases, you would perform polynomial long division to find the equation of the oblique asymptote.
Rational Functions are functions represented by the quotient of two polynomials. The general form is $R(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials.
Degrees of Polynomials are determined by the highest power of $x$ in the polynomial. For example, in $ax^n$, $n$ is the degree of the polynomial.
Undefined Expressions in the context of rational functions are those that result in division by zero, which is not allowed in mathematics.
