Let
f(x)=(4x^2 -7x +2)/(x^2 -16x +63)
Find the equations of the horizontal asymptotes and the vertical asymptotes of f(x). If there are no asymptotes of a given type, enter NONE. If there is more than one asymptote of a given type, give a comma separated list (i.e.: 1, 2,...).
To find the vertical asymptotes, factorize the denominator (if possible) and determine the zeroes of the denominator. Here we find two at x=7 and x=9:
f(x)=(4x^2 -7x +2)/(x^2 -16x +63)
=(4x^2 -7x +2)/((x-7)(x-9))
To find horizontal asymptotes, find the limit of the function as x->∞ or x->-∞.
Lim f(x) x->∞
=Lim (4x²-7x+2)/(x²-16x+63)
=LIm (4x²/x²)
=4
Here we found a horizontal aysmptote at y=4
find the vertical asymptotes of the graph of the function
g(x) = (4+X)/(X^2 (9-XX)
To find the equations of the horizontal asymptotes of the function f(x), we need to examine the behavior of the function as x approaches positive and negative infinity.
First, let's find the horizontal asymptote as x approaches positive infinity (x→+∞).
To do this, we need to compare the degrees of the numerator and denominator.
The degree of the numerator is 2 (highest power of x), and the degree of the denominator is also 2.
Since the degrees of the numerator and denominator are equal, we need to compare the leading coefficients.
The leading coefficient of the numerator is 4, and the leading coefficient of the denominator is 1.
As x approaches positive infinity, the term with the highest power of x (4x^2) will dominate the other terms.
So, the horizontal asymptote as x approaches positive infinity is y = 4/1, which simplifies to y = 4.
Next, let's find the horizontal asymptote as x approaches negative infinity (x→-∞).
Using the same reasoning as before, the leading terms of the numerator and denominator still dominate as x approaches negative infinity.
Therefore, the horizontal asymptote as x approaches negative infinity is also y = 4.
Now, let's find the vertical asymptotes of the function f(x).
To find the vertical asymptotes, we need to determine the values of x that make the denominator equal to zero, since dividing by zero is undefined.
The denominator of f(x) is x^2 - 16x + 63.
To factor the denominator, let's find the values of x that satisfy the equation x^2 - 16x + 63 = 0.
We can factor the equation as (x - 7)(x - 9) = 0.
Therefore, the values of x that make the denominator equal to zero are x = 7 and x = 9.
Finally, the equations of the horizontal asymptotes are y = 4 for both x→+∞ and x→-∞, and the equations of the vertical asymptotes are x = 7 and x = 9.
To find the equations of the horizontal and vertical asymptotes of the function f(x) = (4x^2 - 7x + 2)/(x^2 - 16x + 63), we need to analyze the behavior of the function as x approaches positive and negative infinity.
First, let's consider the horizontal asymptotes. To find them, we need to compare the degrees of the numerator (4x^2 - 7x + 2) and the denominator (x^2 - 16x + 63) of the function.
The degree of the numerator is 2 (highest power of x is x^2) and the degree of the denominator is also 2 (highest power of x is x^2). Since the degrees are the same, we need to compare the leading coefficients (the coefficients of the highest powers of x).
The leading coefficient of the numerator is 4 and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is determined by the ratio of these coefficients, which is 4/1 = 4.
So the equation of the horizontal asymptote is y = 4.
Next, let's consider the vertical asymptotes. To find them, we need to identify any values of x that make the denominator (x^2 - 16x + 63) equal to zero.
We can factor the denominator (x^2 - 16x + 63) by finding two numbers that multiply to 63 and add up to -16. These numbers are -7 and -9.
(x^2 - 16x + 63) = (x - 7)(x - 9)
Setting this expression equal to zero, we get:
(x - 7)(x - 9) = 0
From this equation, we can see that x = 7 and x = 9 are the values that make the denominator equal to zero.
Therefore, the vertical asymptotes are x = 7 and x = 9.
In summary, the equations of the asymptotes are:
- Horizontal asymptote: y = 4
- Vertical asymptotes: x = 7, x = 9