Proving limit theorems when limit tends to infinity

[Alpharup]

Am using Spivak and he defines limit of a function f
1. As it approaches a point a.
2.As it approaches infinity.
He also defines limit f(x)=∞
x->a

But though in solving exercises, we can see that all the three definitions are consistent with each other, I am not satisfied by the argument that these specially defined limits follow all the property like;
Let lim f(x)=m and lim g(x)=n, then to prove lim f(x)*g(x)= m*n...
x->∞ x->∞ x->∞
Let lim f(x)=m here m is not equal to 0, then to prove lim ( 1/f(x))=(1/m)
x->∞ x→∞


In some calculus(like Kuratowski) and analysis books, I could see that they first talk about series, convergence, divergence and so on...Then they talk about epsilon-delta definition of limit and show that this definition is consistent with the one obtained through sequence and series. So, is learning sequences and series(and some concepts like bounded sequence) necessary for proving such and such theorems?

 

[mfb]

It depends on what you want to prove. If you want to prove properties of a limit for sequences and series, you have to know something about them of course. If you want to study everything with epsilon/delta only, you don't need sequences or series.

 

[Alpharup]

I mainly need the proof for epsilon/delta only...

 

[fresh_42]

I mainly need the proof for epsilon/delta only...

How do you know?

 

[Mark44]

I mainly need the proof for epsilon/delta only...

The definitions for limits are different for $\lim_{x \to a}f(x) = L$, $\lim_{x \to \infty}f(x) = L$,$\lim_{x \to a}f(x) = \infty$, and $\lim_{x \to \infty}f(x) = \infty$.
Both $\epsilon$ and $\delta$ appear in the definition of the first limit, $\delta$ (but not $\epsilon$) appears in the third type, and $\epsilon$ (but not $\delta$) appears in the second type. Neither one appears in the fourth limit type.

 

[Alpharup]

So, do you mean to say that, we can still prove the limit properties(multiplication,division) because all the four definitions involve just slight modification of epsilon and delta?

 

[Mark44]

So, do you mean to say that, we can still prove the limit properties(multiplication,division) because all the four definitions involve just slight modification of epsilon and delta?

What I described is not a slight modification of epsilon and delta. In the definitions of these kinds of limits, some of them don't use epsilon or delta (or both) at all. Your book should show each of these limit definitions.

The definitions can be used to prove the multiplication and division properties, but with the usual restrictions. Namely, they shouldn't be any of the indeterminate forms, such as $[\frac 0 0]$, $[\frac {\infty}{\infty}]$, or $[0 \cdot \infty]$.

 

[Alpharup]

Thank you...got it