Find the vertical asymptotes, if any, of the graph of the rational function. Show your work.
f(x) = (x-4)/(x(x-4))
How would I show my work?
the x-4 cancel out and I am left with 1/x. The vertical asymp. is x= 0
that is correct. There is a hole in the graph at x=4, because there
f(x) = 0/0
which is undefined.
f(x) = 1/x everywhere except at x=4.
To find the vertical asymptotes of a rational function, we need to examine the behavior of the function as x approaches positive and negative infinity.
In the given function f(x) = (x-4)/(x(x-4)), we can see that there is a common factor of (x-4) in both the numerator and denominator. This means that we can simplify the function by canceling out this common factor:
f(x) = 1/x
Now, let's analyze the behavior of this simplified function as x approaches positive and negative infinity.
As x approaches positive infinity (x → +∞), the value of 1/x becomes smaller and smaller, tending towards zero. Therefore, the function f(x) approaches zero as x approaches positive infinity: f(x) → 0.
Similarly, as x approaches negative infinity (x → -∞), the value of 1/x also tends towards zero. Thus, the function f(x) approaches zero as x approaches negative infinity: f(x) → 0.
Since the function approaches zero as x goes to both positive and negative infinity, there are no vertical asymptotes in this case. In other words, there are no vertical lines x = a where the function approaches positive or negative infinity.
Hence, in this specific rational function f(x) = (x-4)/(x(x-4)), there are no vertical asymptotes.