Solve Mistake in Limits Problem: Find a & b for Equation Satisfaction

[WiFO215]

I was given this review problem in limits.

Lim_{X --> inf.} (1 - X + X²)^1/2 - aX - b = 0.

I was asked to find a and b such that the above equation is satisfied, which I did as follows:

I removed an X from the entire thing and expanded the term in the brackets using binomial theorem. I get a = 1 and b = -1/2 which is the answer given in the text.

But here's what is bothering me. I did the sum another way and got a different answer and can't put my finger on the mistake.
I remove an X from the root term. And I split the limit across the functions since

Lim _X-->Y f(x) + g(x) = Lim _X-->Y f(x) + Lim _X-->Yg(x)

Lim _X-->Yf(x).g(x) = Lim _X-->Yf(x) . Lim _X-->Y g(x)

This gives me

Lim_{X --> inf.} X(1/X² - 1/X + 1)^1/2 - Lim<sub>X --> inf. (aX + b) = 0

= [LimX --> inf. X] [LimX --> inf.(1/X² - 1/X + 1)^1/2] - LimX --> inf. (aX + b) = 0

Now the term inside the root sign goes to 1. We are left with
LimX --> inf. X - aX - b = 0

This way, the answer is a = 1 and b = 0.

Why am I getting a different answer? I used all the limit rules correctly as far as I can see.</sub>

 

[snipez90]

I think you are haphazardly applying the formulas for the algebraic operations on limits. Remember, you can split limits across functions provided that all limits in question actually exist. For example, I don't think you can split the square root term as you did since X goes off to infinity.

 

[l'Hôpital]

The second way you did it makes it so the 1 - x term disappears. Intuitively, the 1 disappearing is fine since when it comes to infinity, it's meaningless, but the x can't disappear like that.

 

[JG89]

First, $$\lim_{x \rightarrow \infty} x\sqrt[]{(1/x^2 - 1/x + 1)} - ax - b = \lim_{x \rightarrow \infty} x\sqrt[]{(1/x^2 - 1/x + 1)} - (ax + b)$$.

If this limit is to go to 0, then obviously a must be positive (for if a were negative the entire expression we're trying to find the limit of would increase beyond all positive bounds). But if a is positive, then the expression above is of the indeterminate form "infinity - infinity", and so you can't say that the square root expression tends to 1 and the x's will then cancel out.

 

[WiFO215]

Remember, you can split limits across functions provided that all limits in question actually exist. For example, I don't think you can split the square root term as you did since X goes off to infinity.

D'oh! How could I forget that??