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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?