Are you allowed to take for granted that the limit of a product is equal to the product of limits?
hello you all, i'm new here.. i'd like if you help me in this question.
f and g are function defined in a punctured neighborhood of X0, and L is a real number.
assuming lim(x->x0) (f times g)(x)=L
i need to prove\disprove that if lim(x->x0) f(x)=infinity, then lim(x->x0) g(x) exists.
it would be very helpful if you also instruct me in the general way of proving such questions, because i'm kinda new in this subject...
Prove It was asking if you have the theorem that says "if and then . However, that does not appear to me to apply here since does not exist. Instead, it should be apparent that, in order that exist, the limit of g(x) must not only exist but must be a very specific number!
but can we multiple infinity by 0? i mean, if i got you right, you implied that in order that f(x)g(x) doesn't get very large (and to be L, a real number, as it's given), we need something to balance it -> lim g(x). and as i see it, it can only be done if lim g(x) exsits and equal 0, so then lim f(x)g(x) will equal 0 too.
but if i'm not wrong, you can't multiple infinty by 0, can we?