Evaluating a limit

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AmySaunders

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I know I will need to use L'Hopital's Rule, because evaluating this at 0 is indeterminate, but I can't think where to start differentiating.

lim as x approaches zero of (1+2X)^1/x Any ideas, please?
 
AmySaunders said:
I know I will need to use L'Hopital's Rule, because evaluating this at 0 is indeterminate, but I can't think where to start differentiating.
lim as x approaches zero of (1+2X)^1/x Any ideas, please?
Click to expand...

Can you solve \(\displaystyle \displaystyle{\lim _{u \to \infty }}{\left( {1 + \dfrac{2}{u}} \right)^u}~?\)

It is the same problem.
 
AmySaunders said:
I know I will need to use L'Hopital's Rule, because evaluating this at 0 is indeterminate, but I can't think where to start differentiating.

lim as x approaches zero of (1+2X)^1/x Any ideas, please?
Click to expand...

How you solve it depends on what you can use. For example, if you can use the following limit (as pka mentioned)
\(\displaystyle \lim_{n->\infty} (1 + \frac{1}{n})^n\)
you can substitute 1/n = 2x. Another way is if you can use the properties of limits to say if an -> a, then ean -> ea then you could take the log of the sequence and use L'Hopital's Rule. There are undoubtedly other ways to solve the limit also.
 
AmySaunders said:
Actually, the way I work best is if you give me the answer, and I work backwards.
Click to expand...


Totally kidding!

How does one think up substituting 1/x for x and taking the limit at infinity? Is it just lots of experience?

I am also confused at how they are the same thing. I think the limit as x approaches infinity of the problem you suggested would be 1. I was told that the limit of the original problem was e^2, although I haven't yet solved it for myself.

I'm sorry I still don't understand.
 
Your suggestion limit as n approaches infinity of (1+(2/n))^n

is e^2?

as n approaches infinity, the fraction becomes infinitely small, so you basically have 1 to the power of infinity.
 
Last edited:
Okay. I do get it. Thank you for your help!

Still, I would never have thought of it your way by myself. How do I develop my mind so that I can?
 
AmySaunders said:
Your suggestion limit as n approaches infinity of (1+(1/n))^n

is e^2?

as n approaches infinity, the fraction becomes infinitely small, so you basically have 1 to the power of infinity.
Click to expand...
The limit as n goes to infinity of (1+1/n)n is e and is well know mathematically. Basically the answer to your question - Is it just lots of experience? - is yes. Lots of 'practice'

Have you worked out what the expression looks like if you use the substitution suggested, i.e. x = 1/(2n)? Also remember that
lim (an2) is equal to (lim an)2 assuming both limits exist, etc.

Have you tried taking the log of the expression and then using L'Hopital's Rule?
 
Ishuda said:
The limit as n goes to infinity of (1+1/n)n is e and is well know mathematically. Basically the answer to your question - Is it just lots of experience? - is yes. Lots of 'practice'

Have you worked out what the expression looks like if you use the substitution suggested, i.e. x = 1/(2n)? Also remember that
lim (an2) is equal to (lim an)2 assuming both limits exist, etc.

Have you tried taking the log of the expression and then using L'Hopital's Rule?
Click to expand...


Yes, I was able to find the answer I was given once I realized that you don't need to use the quotient rule to differentiate. You can just differentiate the top separately from the bottom. The multiple quotient differentiations were giving me fits! I don't know why it took me so long to realize that, or why I missed it in the first place.
Thank you for your help!
 
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