Challenged in evaluating this limit.

[futb0l]

I really haven't got a clue on how to evaluate this limit. I've tried doing algebraic manipulation, but to no avail. (L'Hopital's rule are not allowed to be used). If someone can give me a hint, that would be great :)

$$\lim_{n\to\infty} \frac{(e^n)^2 + 1}{e^{n^2}+n}$$

 

[Gib Z]

I don't understand your notation on the denominator, is that e^(n^2) or what?

 

[sutupidmath]

I really haven't got a clue on how to evaluate this limit. I've tried doing algebraic manipulation, but to no avail. (L'Hopital's rule are not allowed to be used). If someone can give me a hint, that would be great :)

$$\lim_{n\to\infty} \frac{(e^n)^2 + 1}{e^{n^2}+n}$$

i am not quite sure on this, however i think it should go like this:

substitute e^n=t, than you get, ln e^n=lnt=>n=ln t, when n->infinity, t->infinity

lim(t^2+1)/(t^2+ln t),t->infinity

lim_t->infinity(t^2)/(t^2+lnt)+lim_t->infinity(1)/(t^2+lnt)
the second part is obviously 0 when t-> infinity, so we are left with

lim_t->infinity(t^2)/(t^2+lnt), we know that ln t=ln(1+(t-1)), which is equivalent with t-1, because the limit of their ratio is 1, we substitute it and we get

lim_t->infinity(t^2)/(t^2+t-1), so the limit of this is obviously 1.

 

[Gib Z]

Oh ok I see your latex so I know what you mean. Give me a second to see if I can work it.

 

[sutupidmath]

I don't understand your notation on the denominator, is that e^(n^2) or what?

the denominator i enterpreted as (e^n)^2,
I do not know if this is what he meant?

Is this what you meant?

 

[futb0l]

I meant e^(n^2). So yeah, obviously the one you did with the 't' substitution is not what I'm looking for.

 

[sutupidmath]

than i guess the limit of that is going to be 0, since the denominator growes much faster than the numerator.
I will see if i can work it out now

 

[sutupidmath]

I meant e^(n^2). So yeah, obviously the one you did with the 't' substitution is not what I'm looking for.

try it with the same substitution e^n=t, but just work it out in the form u meant. e^(ln^2 t), take this substitution then for e^(n^2), and i think you will get what u are looking for!

 

[futb0l]

try it with the same substitution e^n=t, but just work it out in the form u meant. e^(ln t)^2, take this substitution then

Ok, so then you get:

$$\lim_{t\to\infty} \frac{t^2 + 1}{e^{(\ln{t})^2} + \ln{t}}$$

So basically you will get infinity at the numerator and denominator, which doesn't lead to anything as far as I can see... *sigh*

 

[theperthvan]

Denominator grows WAY faster than the numerator, so in this case it goes to 0

 

[futb0l]

Can you just say that without doing any calculations?

 

[sutupidmath]

than i guess the limit of that is going to be 0, since the denominator growes much faster than the numerator.
I will see if i can work it out now

i guess we cannot just say this, if this would be in an exam. however it is a clue to lead us on the right direction. i think we should do some calculations.

Although my approach i guess is obviously not appropriate(to much calculations).

 

[futb0l]

I think I just worked it out...

$$\lim_{n\to\infty} \frac{ e^{2n} + 1 }{ e^{n^2} + n }$$

$$\lim_{n\to\infty}\frac{ e^{2n}(1 + e^{-2n}) }{ e^{n^2}(1 + ne^{-n^2}) }$$

$$\lim_{n\to\infty} e^{2n-n^2} \frac {(1 + e^{-2n}) }{(1 + ne^{-n^2}) }$$

$$\lim_{n\to\infty} e^{2n-n^2} * 1$$

So that just equals to 0.

 

[sutupidmath]

$$\lim_{n\to\infty} e^{2n-n^2} \frac {(1 + e^{-2n}) }{(1 + ne^{-n^2}) }$$

$$\lim_{n\to\infty} e^{2n-n^2} * 1$$

So that just equals to 0.

how do you know that the limit of the second part is 1

 

[futb0l]

Yeah, the ne^(-n^2) part is actually kind of part of the question. In the question it gives you limits of various functions - I didn't list it here because there is too many. So yeah, basically that term goes to 0 as n approaches infinite.