a) (2n^2+n+sqrt(n)-3)/(n^2 -5n+7) as n tends to infinity
b) (1+2+3+...+n)/n^2 as n tends to infinity
and you should be able to see now what the limit is....
Always multiply by 1 in the special form of 1 over the highest term of the fraction, in fact, this idea leads to a rule about limits to infinity
The highest power term always dominates, so if its on top then we get infinity, if it's on bottom we get 0, if the degree of top and bottom are equal, then the limit is the ratio of coefficients between the highest degree terms as is the case here
If you want to see this argument rigorously, let me know
Alright, let's only deal with polynomials here...
Say we have
Now we might as well assume that for all other since if it isn't, we just rearrange the expression on top to make it that way.... same on the bottom
So now, 1 of 3 things can happen, either or
So case 1, what if ?
Notice since we assume the cancellation happens on both the top and bottom. Also, since we assume both and are the largest exponents, everything left has a negative exponent, which means it is of the form where and are just 2 numbers. So the limit as n goes to infinity is zero in all those terms. So the limit is the only thing left which is
Now let's assume
Then we do the same thing but we multiply top and bottom by
This will cancel everything on the bottom or give it a negative exponent and since at least the first term of the numerator will still have a positive exponent, so we have something of the form
Which clearly goes to infinity
Now let's assume
Then we do the same thing but we multiply top and bottom by
So everything on the top cancels or has a negative exponent, and at least the first term of the denominator has a positive exponent so we have something of the form which clearly goes to zero
I can spell it fully out like the first example if needed, but I think it should be clear