how do i solve for the limit as x aproaches 3 from the right?

Question

(x^2 - 4x +3) / (root of x-3)

Answer 1

The limit is 0. Here is how that limit is found:

Write the function as f(x) = (x - 3) (x - 1) / [sqrt (x - 3], that is

f(x) = [sqrt (x - 3)] (x -1).

Now put x = 3 + z, where z ---> 0 from the positive side. Then the function takes the form:

g(z) = [sqrt (z)](2 + z), where z = x - 3.

In the limit, as z ---> 0, this ultimately behaves like 2 sqrt (z), since the ' z ' inside the parentheses becomes negligible compared with the ' 2 ' that is also there.

So, the whole function f(x) = g(z) (with x = 3 + z) behaves like
2 sqrt (z) as (positive) z ---> 0, i.e. as x ---> 3 from the right.

The limit, f(x = 3+) is 0 (= zero), but it's approached in an interesting way. The (positive) value reduces steeply and ultimately vertically in the limit as x = 3 (and there, f(x) = 0) is approached from the right.

Live long and prosper.

Answer 2

Since we have an obvious problem at x=3, the equation becomes indeterminate, we must simplify. First factor the numerator into (x-3)(x-1). Then to take care of the denominator, multiply the equation by ((x-3)^(1/2))/((x-3)^(1/2)), (rad (x-3) / (rad (x-3)), this allows you to cancel out some terms and make this more reasonable.
You are now left with the limit as x>3+ of (x-1)(rad(x-3). This gives you one option of solving, if you are able to graph this equation, you can use the picture to tell you the limit. However this way is rather crude and is not thought of as fully using calculus. Another way is to plug in a number slightly larger than 3, let's say 3.01, since you are approaching it from the right and the right is positive. When you take the limit as x approaches 3.01, you will see that it comes very close to 0, so you can tell that will be your answer. I think there is a third way using a table of signs, but I am not fully sure of how to use that method, these two ways will get you the answer, but I am not sure that they will use enough calculus, I hope this helps. If it did, don't forget to pick it as the best answer!! :p

Answer 3

lim [(x^2 - 4x + 3) / sqrt(x - 3)]
x -> 3+

First, factor the numerator.

lim [(x - 3)(x - 1)] / [sqrt(x - 3)]
x -> 3+

Next, recognize that sqrt(x - 3) is the same as (x - 3)^(1/2)

lim [(x - 3)(x - 1)] / [(x - 3)^(1/2)]
x -> 3+

Cancel the (x - 3)^(1/2) in the denominator with 1/2 of a power on the top,

lim [(x - 3)^(1/2) (x - 1)]
x -> 3+

And now, we can safely plug in x = 3 into the limit, so this equals

(3 - 3)^(1/2) (3 - 1)
0^(1/2) (2)
0(2)

= 0

Answer 4

Well you could simplify this to (x-1)*rt(x-3) with cancelling but there is no need. As the x value gets closer and closer to three.. the graph of the function gets closer to zero, until it eventually cannot get any close at x=3. The answer s 0.

iron duke did not take into account the root btw.

Answer 5

(x^2 - 4x +3) / √(x - 3) = (x - 1)√(x - 3)
lim((x - 1)√(x - 3)) = 0
x->3

Answer 6

= lim x--->3 (x-3)(x-1)/(x-3)^.5
=lim x---> 3 (x-1)(x-3)(x-3)^.5/(x-3)
= lim x--> 3 (x-1)(x-3)^.5
= (3-1)(3-3)^.5 = 0