f(x)=(x^3-27)/(x^2-3x)
I need to find the slant.
If the vertical asymptotes is x=0
then is the slant y=x+3?
f(x)=(x-3)(x+3)/x(x-3)= (x+3)/x
The vertical asymptote is indeed x=0
and the slanting asymptote is indeed the line y=x+3
f(x)=(x-3)(x^2+3x+9)/x(x-3)=
(x^2+3x+9)/x=x+3+9/x
If x->inf then 9/x->0
To find the slant of the rational function f(x) = (x^3 - 27)/(x^2 - 3x), we need to perform polynomial long division.
First, divide x^3 by x^2 to get x. Write the result above the line.
x
-----------
x^2 - 3x
Multiply x by (x^2 - 3x) to get x^3 - 3x^2.
x + ?
--------------
x^2 - 3x
Subtract x^3 - 3x^2 from x^3 to get 3x^2.
3x^2
x + ?
--------------
x^2 - 3x
Next, divide 3x^2 by x^2 to get 3. Write this above the line.
x + 3
--------------
x^2 - 3x
Multiply 3 by (x^2 - 3x) to get 3x^2 - 9x.
x + 3
--------------
x^2 - 3x
Subtract 3x^2 - 9x from 3x^2 to get 9x.
9x
x + 3
--------------
x^2 - 3x
We don't have any remaining terms to divide, so the quotient is x + 3.
Hence, the slant of the function f(x) = (x^3 - 27)/(x^2 - 3x) is y = x + 3. Therefore, your observation that the slant is y = x + 3 is correct.
To find the slant or oblique asymptote of the function f(x), follow these steps:
Step 1: Determine if the function has a slant asymptote.
A function can have a slant asymptote if the degree of the numerator is exactly one greater than the degree of the denominator. In this case, the degree of the numerator is 3 (highest power of x) and the degree of the denominator is 2. Since the difference is 1, the function may have a slant asymptote.
Step 2: Find the slant asymptote.
To find the equation of the slant asymptote, perform long division or synthetic division between the numerator and the denominator.
You can use synthetic division in this case:
Write the function as (x^3 - 27)/(x^2 - 3x) = (x^3 - 3x^2 + 3x^2 - 9x + 6x - 18)/(x^2 - 3x).
Perform synthetic division, dividing (x^3 - 3x^2 + 3x^2 - 9x + 6x - 18) by (x^2 - 3x). The quotient will be the equation of the slant asymptote.
x - 6
--------------
x^2 - 3x | x^3 - 3x^2 + 3x^2 - 9x + 6x - 18
x^3 - 3x^2
----------------
0 + 0x - 9x + 6x - 18
- 9x + 18
--------------
0
The quotient is x - 6. Therefore, the slant asymptote of the function is y = x - 6, not y = x + 3.
Lastly, the vertical asymptote x = 0 does not affect the slant asymptote. Vertical asymptotes represent values where the function approaches positive or negative infinity, while a slant asymptote represents how the function behaves for large values of x.
To summarize, the slant asymptote of the function f(x) = (x^3 - 27)/(x^2 - 3x) is y = x - 6.