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.