Find the Asymptotes f(x)=1/(x^2-25)
The given problem is requesting to identify the asymptotes of the function f(x) = 1/(x^2 - 25). To clarify, an asymptote of a function is a line that the graph of the function approaches but never actually reaches as the values of x either increase indefinitely (positive infinity) or decrease without bound (negative infinity). Asymptotes can be vertical, horizontal, or oblique (slant).
For vertical asymptotes, the task involves determining the values of x for which the denominator of the function equals zero, since vertical asymptotes typically occur at the values of x that cause the denominator of a rational function to be zero, which results in the function heading towards infinity.
For the horizontal asymptotes, you would typically calculate the limit of the function as x approaches positive or negative infinity, and these horizontal lines represent the value the function is approaching as the input gets very large or very small.
In this particular case, the goal is to find both vertical and possibly horizontal asymptotes (if any exist) of the given rational function.
$f \left(\right. x \left.\right) = \frac{1}{x^{2} - 25}$
Answer
Solution:
Step 1:
Identify the values for which the function $f(x) = \frac{1}{x^2 - 25}$ is not defined. These are $x = -5$ and $x = 5$.
Step 2:
Examine the limit of $f(x)$ as $x$ approaches $-5$ from the left and from the right. The function approaches positive infinity from the left and negative infinity from the right, indicating a vertical asymptote at $x = -5$.
Step 3:
Examine the limit of $f(x)$ as $x$ approaches $5$ from the left and from the right. The function approaches negative infinity from the left and positive infinity from the right, indicating a vertical asymptote at $x = 5$.
Step 4:
Compile a list of vertical asymptotes for the function: $x = -5$ and $x = 5$.
Step 5:
Review the rules for determining horizontal asymptotes for 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.
Step 6:
Calculate the degrees $n$ and $m$ for the function $f(x)$. Here, $n = 0$ and $m = 2$.
Step 7:
Since the degree of the numerator $n$ is less than the degree of the denominator $m$, the horizontal asymptote is the x-axis, given by $y = 0$.
Step 8:
Determine that there are no oblique asymptotes, as the degree of the numerator is not greater than the degree of the denominator.
Step 9:
Present the complete set of asymptotes for the function:
Vertical Asymptotes: $x = -5$, $x = 5$ Horizontal Asymptote: $y = 0$ No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a function, you need to consider the behavior of the function as $x$ approaches certain critical values. Here are some relevant knowledge points:
Vertical Asymptotes: These occur at values of $x$ where the function becomes undefined and the limits of the function approach infinity (positive or negative) as $x$ approaches these values from either side. To find vertical asymptotes, set the denominator equal to zero and solve for $x$.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator and the denominator of the function. If the degree of the numerator ($n$) is less than the degree of the denominator ($m$), the horizontal asymptote is $y = 0$. If $n = m$, 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: 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. To find an oblique asymptote, you perform long division of the numerator by the denominator and take the quotient (ignoring the remainder).
Limits: The concept of limits is crucial in finding asymptotes. It involves understanding the behavior of a function as the input approaches a certain value, but does not necessarily equal that value.
Rational Functions: A rational function is a function that can be expressed as the ratio of two polynomials. The asymptotic behavior of rational functions is often determined by the degrees and coefficients of these polynomials.
