Hey! I was doing some practice problems and I stumbled upon two rather challenging questions that I cannot for some reason solve.

The questions correspond to letter F and G of number 3)

like i don't even know how to algebraically prove this. I had no problem with the other questions, but I am struggling with these two questions.

Note: My teacher said you cannot use derivatives in order to find the answer. so if possible do not use derivatives.

if possible please show me algebraically or through reasoning why the limit is what it is based on the question.

Thx in Advance