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

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In summary, the conversation discusses a practice question where the answer is e and the reason for this is not given. The question involves an undefined variable z, but it should be replaced by x. The conversation also mentions another definition for e, which involves a limit as x tends to 0. The Taylor series for e is also mentioned as a way to prove the answer.
  • #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
seasponges said:
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).
 
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Thankyou very much!
 

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

What does "(1+x)^(1/z) - Limit as x approaches 0" mean?

The expression "(1+x)^(1/z) - Limit as x approaches 0" represents the limit of the function (1+x)^(1/z) as x approaches 0. This means we are interested in the value that the function approaches as x gets closer and closer to 0.

How do you calculate the limit of "(1+x)^(1/z) - Limit as x approaches 0"?

To calculate the limit of "(1+x)^(1/z) - Limit as x approaches 0", we substitute 0 for x in the expression (1+x)^(1/z) and simplify the resulting expression as much as possible. This will give us the value that the function approaches as x approaches 0.

What is the significance of the limit as x approaches 0 in "(1+x)^(1/z) - Limit as x approaches 0"?

The limit as x approaches 0 in the expression "(1+x)^(1/z) - Limit as x approaches 0" is important because it tells us about the behavior of the function as x gets closer and closer to 0. It can help us determine if the function has a defined value at x=0 or if it approaches infinity or negative infinity.

Can the limit of "(1+x)^(1/z) - Limit as x approaches 0" be evaluated at values other than 0?

Yes, the limit of "(1+x)^(1/z) - Limit as x approaches 0" can be evaluated at any value of x. The limit as x approaches 0 specifically tells us about the behavior of the function at x=0, but we can also evaluate the limit at other values to understand the behavior of the function at those points.

How is the limit of "(1+x)^(1/z) - Limit as x approaches 0" related to the derivative of the function?

The limit of "(1+x)^(1/z) - Limit as x approaches 0" is related to the derivative of the function through the definition of the derivative. The derivative of a function is equal to the limit of the function as the change in x approaches 0. In this case, the limit as x approaches 0 gives us information about the behavior of the function as x gets closer to 0, which is important in determining the derivative of the function at that point.

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