Also, my lecturer prefers to write it like this: $$\lim_{x\rightarrow \infty} \arctan(\ln x) = \begin{bmatrix} t=\ln x \\ x \rightarrow \infty \Rightarrow t \rightarrow \infty \end{bmatrix} \lim_{t\rightarrow \infty} \arctan(t) = \frac{\pi}{2}.$$
$$\lim_{x\rightarrow \infty} \arctan(\ln x) = \begin{bmatrix} t=\ln x \\ x \rightarrow \infty \Rightarrow t \rightarrow \infty \end{bmatrix} = \lim_{t\rightarrow \infty} \arctan(t) = \frac{\pi}{2}.$$ -- Why is it allowed to use equality between these steps, why isn't implication/equivalence used?
As soon as I apply some theorem or operator that changes the range/output of my statement(?), that's when I should use implication/equivalence to show that the application of the operator/theorem yields this new expression that is true?
Okay, so back to my original questions; which operators do I need to look out for that changes the range. Ooh.. I think I just answered my own question!
