limit approaches infinity square root

Question

So the question is:
lim x-->infinity of (sq rt of (x^2 -1))/(2x+1)

First of all, I know we have to use Lhopital's rule. However, I just don't know how.

Second of all, I thought in the end we would get just one value. HOWEVER, my teacher started saying that we would get two and therefore the limit wouldn't exist.
So, I graphed the function on my calculator and noticed that as x approached infinity, it approached just one value (to the right). There was another graph to the left that approached the same, but negative, value as x approached NEGATIVE infinity.

Well, wait a minute, I said...why are we counting the two values if this problem is asking when x--> infinity (positive infinity, I assumed, since it didn't have a negative sign).

This is what he said: Oh, because in saying x approaches infinity we mean x approaches both directions of infinity (negative and positive). This is because when talking about infinity we don't do "approaching from the left/ approaching from the right of positive or negative infinity...it's just negative or positive infinity" .

I know, it makes no sense. So, that is why, my friends...I am completely confused and have a major headache because I just want to understand and, obviously, I am far from understanding. I thought it's just positive or negative infinity, not 'infinity as a whole'. what does that even mean??


Can someone explain why this limit approaches two different values and therefore doesn't exist, as my teacher so confusingly put it?

Answer 1

Limits x tends to infinite Sqrt(x^2-1)/(2x+1)

 

We have to apply L' hopital's rule

 

Taking derivative of num and den

 

0.5((x^2-1)^(-1/2))*2x/2

 

=x/2*Sqrt(x^2-1)

 

=0.5/sqrt(1-(1/x^2))

 

Now taking limits x tends to infinite

 

=1/2

Answer 2

According to the rules of radicals, (√(x² - 1)) / (2x + 1) can be rewritten as

 

√((x² - 1) / (2x + 1)(2x + 1)) = √((x² - 1) / (4x² + 4x + 1))

 

Now we can take the limit of this function as x approaches infinity.  To do this, we divide the numerator and denominator by the highest power of x in the denominator.

 

lim             √((x² - 1) / x² / (4x² + 4x + 1) / x²)

x -->∞

 

 

 

lim          √((1 - 1/x²) / (4 + 1/x  + 1/x²))

x -->∞

 

 

We know that a small number divide by a very very very large number is close to zero.

 

lim     √(1 / 4)

x -->∞

 

 

Since a negative number squared and a positive number squared gives us a positive number,

 

The limit as x reaches infinity is -1/2 and 1/2.

 

-1/2    and     1/2     are also our horizontal asymptotes.