Hi. I need help in this problem. If someone could help that will be very nice. Sorry but I don't know how to write math equations on a forum post.
Any help would be nice. Thanks
1. Knowing that f and g are two functions and:
lim f(x)= +oo (+infinite) and lim g(x)=0
x->a x->a
(a) Show that choosing conveniently the functions f and g and defining h=f*g we can have the follow situations:
(i) lim h(x) = +oo
x->a
(ii) lim h(x) = -oo
x->a
(iii) lim h(x) = 0
x->a
(iv) lim h(x) = 5
x->a
(v) lim h(x) = -5
x->a
(vi) lim h(x) do not exist , without being +oo or -oo
x->a
Note: The point a € R is a generic point.
€=belong
assuming you actually meant
, and(i) lim h(x) = +oo
x->a
, and
(ii) lim h(x) = -oo
x->a
, and
(iii) lim h(x) = 0
x->a
, and(iv) lim h(x) = 5
x->a
, and(v) lim h(x) = -5
x->a
, and(vi) lim h(x) do not exist , without being +oo or -oo
x->a
(by the way, we can choose a right? if you're not allowed to, i leave it to you to modify my responses)
it's kind of hard to explain, you just have to "see" it
for instance, the first. i wanted ,
since i want to get i said to myself. how can i get the limit to go to infinty? well, i can "divide by zero" or i can have the function be, say x, or e^x or whatever such that when x goes to infinity, the function goes to infinity, so i chose the latter.
so ok, obviously now i want , so i choose
now i need to get and
well, i know that because i have seen that limit so many times (and you should have too). so that's my , now, what do i want to be?
i know , so could that be ? well, no, since i would have which does not go to infinity as x goes to infinity. so i realize, i just need one more factor of x to get my , so i made . so now, , and as desired.
similar reasoning went into the others. just try to think of well-known limits and tweek them to fulfill the conditions. as you see, i did not use any complicated functions (well, the last one was kind of complicated, but only slightly), just keep it simple.