Basic Algebra Concept Involving Calculus

[mikagurl]

Hi all. I have a basic question that I cannot grasp the concept of. I am studying limits.

Suppose I have to evaluate the limit, as x approaches positive infinity, of (2x^4-x^2+8x) over -5x^4+7.

To solve this problem I would factor out the largest power of x that is in the denominator which is x^4. In doing so, I would end up with

x^4 (2-(1/x^2)+(8/x^3) over x^4 (-5+(7/x^4)

I totally understand the factorization that I just explained.

This is the concept that I do not understand. So again, suppose I have to evaluate the limit, as x approaches positive infinity, of

(e^10x - 4e^6x + 3e^x + 2e^-2x - 9e^-15)

I would again start by factoring the largest power of x which is e^10x. However on this one why am I bringing the exponents to the numerator? In the first example I did not bring my exponents to the numerator. For an example if I am factoring the e^10x from the second term it should be 4/e^4x. However, I have 4e^-4x. I hope everyone can understand my question. I do not know how to use La Tex yet. Thanks

 

[Ishuda]

It is just a matter of how you like to see the expression written (or how your teacher likes to see it written). 4/e^x and 4 e^-x are generally acceptable as being the same quantity unless it is specifically asked for one way. For example one might ask the expression be written with no negative exponents. In that case you would write it the first way. As far as the first expression goes, you also could have written 7 x^(-4) instead of 7/x^4 and it would have been acceptable unless you had been asked to write it differently.