So the example I have for context is limx-->infinity sqrt(x+3x^2)/4x-1
Why is it that it isn't as simple as just finding the value of x that would make the denominator 0?
Additionally, why isn't the first step to divide the numerator and denominator into two separate limit problems and then use each of their outputs for the numerator and denominator in my answer, respectively?
Just wondering out of curiosity, I'm.
Thanks
First, equating the denominator, which is 4x - 1, to zero means setting x = 1/4? Why do you think that 1/4 is approaching infinity?
Second, one of the general theorems of limits is this:
[MATH]\lim_{x \rightarrow a}f(x), \ \lim_{x \rightarrow a}g(x) \in \mathbb R \text { and} \lim_{x \rightarrow a}g(x) \ne 0 \implies \lim_{x \rightarrow a} \dfrac{f(x)}{g(x)} = \dfrac{\displaystyle \lim_{x \rightarrow a}f(x)}{\displaystyle \lim_{x \rightarrow a}g(x)}.[/MATH]
Your thought that the limit of a quotient equals the quotient of the limits is generally true, but there are exceptions. This problem
potentially involves one of the exceptions so you cannot directly use this theorem. The limits in this case are infinity and so not real numbers.
You are being introduced to enough standard analysis to persuade you that calculus is not mumbo jumbo. Personally I find standard analysis incredibly ugly and hope that someday mathematicians find a more elegant way to deal with the problems of
[MATH]\dfrac{0}{0},\ \dfrac{\infty}{\infty}, \text { and } 0 + 0 +\ ...[/MATH]
But right now what we have is standard or non-standard analysis, and most schools use standard analysis.
As a practical matter, most problems of limits involve using the general laws of limits and finding equivalent descriptions of functions that cannot be dealt with
directly using those general laws. That is what pka showed you how to do in this case.
