Lim n[tex]\rightarrow[/tex][tex]\infty[/tex] (1+(x/n))[tex]^{n}[/tex]

  • Thread starter torasstripes
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In summary, the conversation discusses using L'Hopital's rule to find the limit as n goes to infinity of an exponential function with a form of "0/0". The strategy involves taking the logarithm of the function and rewriting it as a quotient before applying L'Hopital's rule.
  • #1
torasstripes
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lim n[tex]\rightarrow[/tex][tex]\infty[/tex] (1+(x/n))[tex]^{n}[/tex]

I have no idea where to start. Can anyone who knows what to do give me a hint or tell me the first step? Thanks
 
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  • #2
That looks an awful lot like an exponential function to me. Take the logarithm of [itex](1+(x/n))^n[/itex] and you have a limit, as n goes to infinity, of the form [itex]0*\infty[/itex]. That can be rewritten as so that it is of the form "0/0" and, even though here n is an integer, you can use L'Hopital's rule.
 
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  • #3
HallsofIvy said:
That looks an awful lot like an exponential function to me. Take the logarithm of [itex]1+(x/n))^n and you have a limit, as n goes to infinity, of the form [itex]0*\infty[/itex]. That can be rewritten as so that it is of the form "0/0" and, even though here n is an integer, you can use L'Hopital's rule.

I'm pretty sure I can't pull the one out of the parenthesis like that. Like when you foil you always have a middle term? So with this we should have n+1 terms, but if we take the one out it chages the equation and we only have two terms if we expand it
 
  • #4
Maybe I'm wrong. If I can't come up with something better I will do that
 
  • #5
The 1 was supposed to be in the parentheses! I have edited my previous post.

The logarithm of [itex](1+ (x/n))^n[/itex] is n log(1+ (x/n)) which, as I said, is of the form "[itex]0*\infty[/itex]". You can write that as log(1+ (x/n))/(1/n) so that it is now of the form "0/0". Apply L'Hopital's rule treating the n as a continuous variable.
 
  • #6
alright. thanks so much!
 

1. What does the expression (1+(x/n))^n represent?

The expression (1+(x/n))^n represents the limit as n approaches infinity of the function (1+(x/n))^n. This means that as n gets larger and larger, the value of the function will approach a certain number.

2. How do you calculate the limit of (1+(x/n))^n as n approaches infinity?

To calculate the limit of (1+(x/n))^n as n approaches infinity, you can use the formula lim n[tex]\rightarrow[/tex][tex]\infty[/tex] (1+(x/n))[tex]^{n}[/tex] = e^x. This means that the limit is equal to the value of the exponential function e^x, where x is the value inside the parentheses.

3. What is the significance of the limit (1+(x/n))^n as n approaches infinity?

The limit (1+(x/n))^n as n approaches infinity is significant because it represents the behavior of the function as the input value (n) gets larger and larger. This is known as the limit at infinity and can help us understand the long-term behavior of a function.

4. Can the limit (1+(x/n))^n be calculated for any value of x?

Yes, the limit (1+(x/n))^n can be calculated for any value of x. This is because the limit is independent of the value of x and only depends on the behavior of the function as n approaches infinity.

5. How does the limit (1+(x/n))^n relate to the concept of exponential growth?

The limit (1+(x/n))^n is closely related to the concept of exponential growth. As n approaches infinity, the value of the function (1+(x/n))^n will approach e^x, which is the value of exponential growth. This means that the function will grow exponentially as n gets larger, which is a common pattern in many real-world scenarios.

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