Limit Help: (x - 1)/(sqrt(x) - 1) as x -> 1

[ladyfire]

1) I need to find the limit, as x approaches 1, of this expression:

. . .(x-1) / (sqrt(x) - 1)

(Only the x is under the square root)

I know that I need to get the radical out of the denominator, so I multiply both the numerator and the denomenator by the conjugate, which is (sqrt(x) + 1). Now the denomenator is (x - 1), but what is the numerator? I have forgotten how to multiply them.

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Also is this correct?

2) the limit, as x approaches 3, of (x^2 - 9) / (x^2 - 4x = 3) is 1.5?

 

[stapel]

1) You're on the right track, but did you notice that you still have the division-by-zero problem? So the trick is to work backwards. Instead of creating a difference of squares in the denominator by multiplying by the conjugate, "find" a difference of squares in the numerator, factor, and cancel:

. . . . .x - 1 = (sqrt[x])² - 1² = (sqrt[x] - 1)(sqrt[x] + 1)

Then:

. . . . .(x - 1)/(sqrt[x] - 1)

. . . . .= [(sqrt[x] - 1)(sqrt[x] + 1)] / (sqrt[x] - 1)

I'll bet you can see the answer now.... :wink:

2) There appears to be a typo in the denominator...?

Eliz.

 

[ladyfire]

Sorry the = should be a +

thanks for the help. I think I got the 1rst one. I got the limit being 2.

 

[stapel]

1) Yes.

2) How are you getting your limit value? (I factored, cancelled, and evaluated, and got a different value, is why I ask.)

Eliz.

 

[ladyfire]

I facotred like this


(x-3)(x+3) /
(x-3)(x-1)

therefore the (x-3) cancel out and I am left with (x+3) /(x-1)
then I just plugged in 3 and got 6/4 or 1.5

 

[stapel]

But 3 - 1 is not 4. :wink:

Eliz.