SOLUTION: I whant to calcule the limit of this equation wen it tends to +infinit lim x--> +oo(2x-1)/((4x^2+1)^1/2-(x^2-1)^1/2)

Question 25336: I whant to calcule the limit of this equation wen it tends to +infinit
lim x--> +oo(2x-1)/((4x^2+1)^1/2-(x^2-1)^1/2)
Answer by kev82(151) (Show Source): You can put this solution on YOUR website!
Hi,
The trick when dealing with infinite limits, is to rearrange it so it's made of functions that get smaller as x gets large. Functions such as

are quite good. So we've got

The more maths you do you get an intuition for things link this, and you can only really get it by practice. Here's how I suggest we proceed - We could make the numerator nice by dividing it by x. To keep the same function we would also have to divide the denominator by x, so let's try that and see what happens.
The numerator becomes:

Unfortunately, the denominator is a bit trickier, but using the fact that

for large posotive x, the denominator becomes:

Tidying up, we get:

Now notice that x only appears as

or nicer, which all go to zero as x get's big. So the limit evaluates to:

Hope that helps,
Kev
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