rational functions--> (1) / (x-1) ^ 2?

Question

how do i find...
-horizontal asymptote, would it be 1 since x-1=0 -->1 ?? and would there only be one answer or two? since it's squared?
-domain, would it be (infinity, 1)hole(1, infinity) assuming the horiz. asy. =1??
-vertical asymptote, how would i find the degree for the numerator when there is no "x" attached to the "1"?
-range

explanations would be appreciated and thank you for your help.

Answer 1

There is a horizontal asymptote at y = 0 since
n < d

There is a vertical asymptote at x = 1 since (1 - 1)^2 = 0

Domain is (-infinity,1)U(1,infinity) since x cannot equal 1

Range is (-infinity,0)U(0,infinity) since y cannot equal zero.

See your other question for more explanations..

Answer 2

Domain:
You have to study when the denominator = 0. It is when x=1, so
domain: all the real numbers except for 1, because the function is not defined in it.

D=R\{1}

What does it happen when x=1?

lim (1) / (x-1) ^ 2= + ∞
x-->1-

lim (1) / (x-1) ^ 2= + ∞
x-->1+

So you have a vertical asymptote: x=1.

lim (1) / (x-1) ^ 2 = 0
x--> - ∞

lim (1) / (x-1) ^ 2 = 0
x--> + ∞

This is a horizontal asymptote: f(x)=y=0 (the x-axis) to the right and to the left.

Range

As you have also a positive number in the numerator, you have to study the sign of the denominator.

The denominator is = 0 only when x = 1, but the function is not defined in 1, so in the other cases the denominator is always positive for every x value, because it is in a square.
So, the range is (0, + ∞), (except for 0), that is: all the positive numbers.

Is it clear enough?

Have a great time with mathematics...see you!

Answer 3

As x is great in the negative or positive direction
the denominator becomes great thus making y(I assuume
it's equal to y) closer and closer to zero, but it's on the
positive side of the x axis since the denom. is a square
and must be positive. Thus the x-axis is a horizontal asymptote. As x from the negative side gets
greater(smaller absolute value) and x from the positive side gets smaller, y starts
to rise dramatically from both sides and approaches a vertical asymptote at x=1, since as x --> 1,
denom. --->0 and y ---> infinity.