Limit question series representation

[Gekko]

Homework Statement

Lim (t->0+) [exp(-x^2/t)] / sqrt(t)

The Attempt at a Solution

Ive looked at series representation:

1/sqrt(t) * [exp(-x^2) - 1+t^2/2 + t^3/6 + ...] but that doesn't help.
L'Hopital's rule isn't any use either because the t terms are in the denominator but I know this tends to 0

 

[Coto]

Try making the substitution:
$$t \rightarrow \frac{1}{u}$$
and taking the limit $$u \rightarrow \infty$$.

Retry L'Hopital's rule.

 

[Mark44]

Homework Statement

Lim (t->0+) [exp(-x^2/t)] / sqrt(t)

The Attempt at a Solution

Ive looked at series representation:

1/sqrt(t) * [exp(-x^2) - 1+t^2/2 + t^3/6 + ...] but that doesn't help.

Where does the - 1+t^2/2 + t^3/6 + ... part come from?

L'Hopital's rule isn't any use either because the t terms are in the denominator but I know this tends to 0

 

[Gekko]

hi Mark44, the series comes from splitting out into exp(-x^2) and exp(t) and then using the series representation of the exponential function on the second term.

The substitution approach works. Thanks Coto

 

[Mark44]

But you seem to be saying that e^(a/b) = e^a - e^b. That's not true.

 

[Gekko]

Sorry you're right. That a mistake

 

[Gekko]

Coto, actual I don't see how that sub helps because I now have:

(u^3/2)(x^2/2)exp(-ux^2/4) as u tends to inf

The exponential term tends to zero however the u^3/2 tends to infinity

 

[Dick]

Go back to exp(-x^2/t)*t^(-1/2). Take the log. Now try to figure out what the limit of the log might be. l'Hopital will help with some steps on the way.

 

[Gekko]

So if we perform l'hopital first and then log we obtain

Ln(L)=ln(-x^2)-ln(2t^0.5)+x^2/4t

The limit of -ln(2t^0.5) is -inf and the limit of x^2/4t is also inf as t -> 0 and exp(inf) = inf which means the limit is indeterminate

 

[Gekko]

Is l'hospital's rule the problem in this situation?

 

[Dick]

Is l'hospital's rule the problem in this situation?

I would do log first. If you get something like (1/t)-log(t) write it as (1-t*log(t))/t and think about that. l'Hopital might help with thinking about the t*log(t) part.

 

[Gekko]

I would do log first. If you get something like (1/t)-log(t) write it as (1-t*log(t))/t and think about that. l'Hopital might help with thinking about the t*log(t) part.

ln(L) = (-x^2-4tln(sqrt(t))) / 4t

L'Hopital's: f'(t)/g'(t) where f(t) = -[x^2+4tln(sqrt(t))] and g(t)=4t

f'(t) = -[4ln(sqrt(t)) + 4t/2x], g'(t) = 4

f'(t)/g'(t) = -[ln(sqrt(t)) + t/2x] = inf. L = e^inf = inf which is not the correct answer

Where is this approach going wrong?

 

[Dick]

ln(L) = (-x^2-4tln(sqrt(t))) / 4t

L'Hopital's: f'(t)/g'(t) where f(t) = -[x^2+4tln(sqrt(t))] and g(t)=4t

f'(t) = -[4ln(sqrt(t)) + 4t/2x], g'(t) = 4

f'(t)/g'(t) = -[ln(sqrt(t)) + t/2x] = inf. L = e^inf = inf which is not the correct answer

Where is this approach going wrong?

Think about the t*log(sqrt(t)) part separately before you apply l'Hopital to the whole thing. It may NOT even be indeterminant. Write it as log(sqrt(t))/(1/t). Apply l'Hopital to just that. It wouldn't hurt to notice log(sqrt(t))=(1/2)*log(t) either.

 

[Gekko]

Think about the t*log(sqrt(t)) part separately before you apply l'Hopital to the whole thing. It may NOT even be indeterminant. Write it as log(sqrt(t))/(1/t). Apply l'Hopital to just that. It wouldn't hurt to notice log(sqrt(t))=(1/2)*log(t) either.

So, just to follow up: log(L) = -[(x^2-4tlog(sqrt(t)))/4t]

applying l'hopital to 4tlog(sqrt(t)) or 2tlog(t) where f(t) = 2log(t) and g(t)=1/t gives

-2t^2/t = -2t

Substituting, our limit becomes log(L) = (-x^2 + 2t)/4t = -x^2/4t + 1/2 = -inf
exp(-inf) = 0.

Thank you!

 

[Dick]

So, just to follow up: log(L) = -[(x^2-4tlog(sqrt(t)))/4t]

applying l'hopital to 4tlog(sqrt(t)) or 2tlog(t) where f(t) = 2log(t) and g(t)=1/t gives

-2t^2/t = -2t

Substituting, our limit becomes log(L) = (-x^2 + 2t)/4t = -x^2/4t + 1/2 = -inf
exp(-inf) = 0.

Thank you!

It might be a little dangerous to be saying things like lim 4tlog(sqrt(t))=(-2t). The limit is a number. Not an expression. I would say the limit is zero and leave it at that.

 

[Gekko]

It might be a little dangerous to be saying things like lim 4tlog(sqrt(t))=(-2t). The limit is a number. Not an expression. I would say the limit is zero and leave it at that.

Ah yes I see your point. Thanks