I am just a beginner in Math and a little confused about the point-wise convergence. I a getting contradicting results between indeterminate form and point-wise convergence. Is it common?
consider for example a sequence arc(tan nx). My book says for x > 0 the limit n-> inf arc (tan nx) converges to pi/2 but what if x->0, how then can we say that the limit exists?
A very trivial question is, what if x->0 and n-> inf, doesn't nx become an indeterminate form then? Why then do we simply say nx tends to infinity and arc(tan nx) -> pi/2. Is this logic? or should I do rote learning in Maths too? Save me.
You should keep in mind that, at any given moment, either you are dealing with $\lim_{n\to\infty}\arctan(nx)$, or with $\lim_{x\to a}\arctan(nx)$. Not with both of them at the same time! And pointwise convergence is about $\lim_{n\to\infty}\arctan(nx)$.