Is Lim (f*g) = lim f * lim g always true if all limits exists?

Steven G

Steven G

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It can be shown that Lim(n-->oo) [(1+1/n)n*n -en] = -e/2 (and I can do this)

Now consider this method: (note all limits are as x goes to infinity)
Lim(n-->oo) [(1+1/n)n*n - en] =

Lim(1+1/n)n * Lim(n) -Lim(e)* Lim(n)

= e*Lim(n) - e*Lim(n)

= e(Lim(n) - Lim(n))

=e(Lim(n-n)) = eLim(0) = e*0 = 0

Where is the mistake??
 
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Jomo said:
It can be shown that Lim(n-->oo) [(1+1/n)n*n -en] = -e/2 (and I can do this)

Now consider this method: (note all limits are as x goes to infinity)
Lim(n-->oo) [(1+1/n)n*n - en] =

Lim(1+1/n)n * Lim(n) -Lim(e)* Lim(n)

= e*Lim(n) - e*Lim(n)

= e(Lim(n) - Lim(n))

=e(Lim(n-n)) = eLim(0) = e*0 = 0

Where is the mistake??
Click to expand...

Do all the limits you used exist?
 
Dr.Peterson said:
Do all the limits you used exist?
Click to expand...
Let's see.
Lim(1+1/n)n = e

Lim(e) = e

Lim(n-n) = Lim(0) = 0.

The 1st two limits are clearly correct. You can do simplification of what you are taking the lim of and that is what I did in the 3rd limit. Where did I go wrong?

Edit: well lim (n) dne
 
Jomo said:
Let's see.
Lim(1+1/n)n = e

Lim(e) = e

Lim(n-n) = Lim(0) = 0.

The 1st two limits are clearly correct. You can do simplification of what you are taking the lim of and that is what I did in the 3rd limit. Where did I go wrong?
Click to expand...

How about lim(n)? You assumed that lim(n) - lim(n) = lim(n-n).
 
Jomo said:
Correct, I did assume that. How about lim[(x+k) - (x)] = lim (k) = k? is this also wrong?
Click to expand...
Are you suggesting that

lim[(x+k) - (x)] = lim (k) = k, while lim (x+k) - lim (x) dne???
 
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Jomo said:
Correct, I did assume that. How about lim[(x+k)- (x)] = lim (k) = k? is this also wrong?
Click to expand...

Of course not; that's just simplification. (I assume you are taking this limit as x-->inf.)

Jomo said:
Are you suggesting that

lim[(x+k) - (x)] = lim (k) = k, while lim (x+k) - lim (x) dne???
Click to expand...

Yes I am, though I would put it differently:

If the individual limits lim(u) and lim(v) do not exist, you can't say that lim(u - v) = lim(u) - lim(v), because the right side doesn't mean anything.

Check the full statements of the properties of limits.

You started out by asking, "Is lim (f*g) = lim f * lim g always true if all limits exist?" That is an appropriate question to have asked; the answer is yes. But you forgot to ask, "Is lim (f-g) = lim f - lim g always true if all limits do not exist?" The answer there is no. That is the point where your work fails: lim(n) does not exist!
 
Dr.Peterson said:
Of course not; that's just simplification. (I assume you are taking this limit as x-->inf.)



Yes I am, though I would put it differently:

If the individual limits lim(u) and lim(v) do not exist, you can't say that lim(u - v) = lim(u) - lim(v), because the right side doesn't mean anything.

Check the full statements of the properties of limits.

You started out by asking, "Is lim (f*g) = lim f * lim g always true if all limits exist?" That is an appropriate question to have asked; the answer is yes. But you forgot to ask, "Is lim (f-g) = lim f - lim g always true if all limits do not exist?" The answer there is no. That is the point where your work fails: lim(n) does not exist!
Click to expand...
It is sad because a dozen years ago I would have known that. I need to start doing math much more than I am.
 
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