(1+x)^(1/z) - Limit as x approaches 0

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  • #1
seasponges
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I was given this as a practice question, and assumed that the answer would be 0.

The answer is e, but no explanation is given and I cannot figure out why this is. z is undefined in the question.

If anyone could shed some light it would be greatly appreciated.
 

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  • #2
Curious3141
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I was given this as a practice question, and assumed that the answer would be 0.

The answer is e, but no explanation is given and I cannot figure out why this is. z is undefined in the question.

If anyone could shed some light it would be greatly appreciated.

There shouldn't be a z, it should simply be replaced by x.

[tex]\lim_{x \rightarrow 0} {(1 + x)}^{\frac{1}{x}} = e[/tex]

That's actually one of the definitions for e (the base of natural logarithms). If you want a fairly elementary (but not very rigorous) way of proving it, consider putting [itex]y = \frac{1}{x}[/itex], from which you get the equivalent definition:

[tex]\lim_{y \rightarrow \infty} {(1 + \frac{1}{y})}^y = e[/tex]

and apply Binomial theorem to the LHS. Expand and consider the limit as y tends to infinity. Now compare that with the Taylor series for e (e1).
 
  • #3
seasponges
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Thankyou very much!
 

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