TITLE: How to place a limit that it's inside the integral, outside.
QUESTION [6 upvotes]: I did this:
$$\int_{1}^t x^{-1}dx=\int_{1}^t\lim_{n\rightarrow -1}{x^n}dx =\lim_{n\rightarrow -1}\int_{1}^t{x^n}dx $$ just to have a way to approximate $\ln t$. $$\ln{t}=\lim_{h\rightarrow 0}\frac{x^{h}-1}{h}$$
The second expression may be correct, but I was told I cannot say $\int_a^b \lim_{n\rightarrow -1}x^{n}dx=\lim_{n\rightarrow -1}\int_a^b x^n dx$ without previously prooved some statements. If so, what are those statement and where can I find information about this?

REPLY [8 votes]: In general, you cannot interchange two limits. In this case, recall that the integration is a actually a limit, for instance, the limit of a Riemann sum. Hence, you need to be careful, in general:
$$\int_a^b \lim_{n \to \infty} f_n(x) dx \neq \lim_{n \to \infty} \int_a^b f_n(x) dx$$
You may want to read through the following posts to figure out under what conditions, you can swap two limits.
When can the order of limit and integral be exchanged?
Can a limit of an integral be moved inside the integral?
When can we exchange order of two limits?