Limit Help: (x - 1)/ Sqare root (x^2 - 1 ) as x -> 1

[Marco]

1) I need to find the limit, as x approaches 1, of this expression
(x-1) / (sqrt(x^2 - 1)

do i need to conjugate the denominator like this?
(x-1) / (sqrt(x^2 - 1) • (sqrt(x^2 + 1) / (sqrt(x^2 + 1) ?

I know that I can conjugate IF the expression is like this :

(x-1)/ (sqrt (x^2) - 1) • (sqrt (x^2) + 1) / (sqrt (x^2) + 1)

but in the 1st expression , I don't know what to do.

 

[Deleted member 4993]

1) I need to find the limit, as x approaches 1, of this expression
(x-1) / (sqrt(x^2 - 1)

do i need to conjugate the denominator like this?
(x-1) / (sqrt(x^2 - 1) • (sqrt(x^2 + 1) / (sqrt(x^2 + 1) ?

I know that I can conjugate IF the expression is like this :

(x-1)/ (sqrt (x^2) - 1) • (sqrt (x^2) + 1) / (sqrt (x^2) + 1)

but in the 1st expression , I don't know what to do.

In this case, you can conjugate - but that won't help. Instead:

(x-1) / sqrt(x^2 - 1) = [(x-1) / sqrt(x^2 - 1)] • sqrt(x^2 - 1) / sqrt(x^2 - 1) = [(x-1) / (x^2 - 1)] • sqrt(x^2 - 1) = [1 / (x + 1)] • sqrt(x^2 - 1)

Now take the limit....

 

[stapel]

1) I need to find the limit, as x approaches 1, of this expression
(x-1) / (sqrt(x^2 - 1)

do i need to conjugate the denominator like this?
(x-1) / (sqrt(x^2 - 1) • (sqrt(x^2 + 1) / (sqrt(x^2 + 1) ?

Note that the innards of the radical are a product; specifically, they're a difference of squares:

. . . . .\(\displaystyle x^2\, -\, 1\, =\, (x\, -\, 1)(x\, +\, 1)\)

This means that the expression can be restated as:

. . . . .\(\displaystyle \dfrac{x\, -\, 1}{\sqrt{\strut (x\, -\, 1)(x\, +\, 1)\,}}\, =\, \dfrac{\left(\sqrt{\strut x\, -\, 1\,}\right)^2}{\sqrt{\strut x\, -\, 1\,}\, \sqrt{\strut x\, +\, 1\,}}\)

Where does this lead?

 

[Marco]

In this case, you can conjugate - but that won't help. Instead:

(x-1) / sqrt(x^2 - 1) = [(x-1) / sqrt(x^2 - 1)] • sqrt(x^2 - 1) / sqrt(x^2 - 1) = [(x-1) / (x^2 - 1)] • sqrt(x^2 - 1) = [1 / (x + 1)] • sqrt(x^2 - 1)

Now take the limit....

So, sqrt (x^2-1) / x+ 1 , how do I remove the radical?

or I will subtitute the x now by 1? , = sqrt (1^2 - 1) / 1+1 = sqrt(0)/ 2 = -> 0/2 <- am I correct? or not?

Thank you for replying

 

[Marco]

Note that the innards of the radical are a product; specifically, they're a difference of squares:

. . . . .\(\displaystyle x^2\, -\, 1\, =\, (x\, -\, 1)(x\, +\, 1)\)

This means that the expression can be restated as:

. . . . .\(\displaystyle \dfrac{x\, -\, 1}{\sqrt{\strut (x\, -\, 1)(x\, +\, 1)\,}}\, =\, \dfrac{\left(\sqrt{\strut x\, -\, 1\,}\right)^2}{\sqrt{\strut x\, -\, 1\,}\, \sqrt{\strut x\, +\, 1\,}}\)

Where does this lead?

x-1/ sqrt (x^2 - 1) = Sqrt(x-1)^2 / (sqrt (x-1) (sqrt(x+2) = sqrt(x-1)/ sqrt (x+2) = [sqrt (x-1) / sqrt (x+2)] • [sqrt (x+2) / sqrt(x+2)] = sqrt(x-1) / x+2 • sqrt(x+2) = ( I don't know what to do next )

Thank you for Replying sir

 

[Deleted member 4993]

So, sqrt (x^2-1) / x+ 1 , how do I remove the radical?

or I will subtitute the x now by 1? , = sqrt (1^2 - 1) / 1+1 = sqrt(0)/ 2 = -> 0/2 <- am I correct? or not?

Thank you for replying

How did you get that?!! Show detailed work.

 

[Marco]

How did you get that?!! Show detailed work.

(x-1) / sqrt(x^2 - 1) = [(x-1) / sqrt(x^2 - 1)] • sqrt(x^2 - 1) / sqrt(x^2 - 1) = [(x-1) / (x^2 - 1)] • sqrt(x^2 - 1) = [1 / (x + 1)] • sqrt(x^2 - 1) = [sqrt (x^2 -1 ) / x + 1 ], am I right sir? correct me sir if I am wrong,

thank you very much for your time

 

[Deleted member 4993]

(x-1) / sqrt(x^2 - 1) = [(x-1) / sqrt(x^2 - 1)] • sqrt(x^2 - 1) / sqrt(x^2 - 1) = [(x-1) / (x^2 - 1)] • sqrt(x^2 - 1) = [1 / (x + 1)] • sqrt(x^2 - 1) = [sqrt (x^2 -1 ) / (x + 1) ], am I right sir? correct me sir if I am wrong,

thank you very much for your time

Must have those parentheses (). Otherwise meaning is changed.

Now take the limit .... what is stopping you?

 

[Marco]

Must have those parentheses (). Otherwise meaning is changed.

Now take the limit .... what is stopping you?

So, sqrt (x^2-1) / (x+1) , how do I remove the radical?

I will subtitute the x by 1 now sir?

sqrt ((1)^2 - 1) / (1+1) = sqrt(0)/ 2 = 0/2 or 0,

the limit of (x-1) / (sqrt(x^2 - 1) as x approaches 1 is 0

am I correct sir?

thank you for your time sir

 

[Deleted member 4993]

So, sqrt (x^2-1) / (x+1) , how do I remove the radical?

I will subtitute the x by 1 now sir?

sqrt ((1)^2 - 1) / (1+1) = sqrt(0)/ 2 = 0/2 or 0,

the limit of (x-1) / (sqrt(x^2 - 1) as x approaches 1 is 0

am I correct sir?

thank you for your time sir

Correct

 

[Marco]

Correct

Thank You Very Much for getting back to me Sir