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glueball8
- 346
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proof lim (x+1)^(1/x)=e. Where can I find the proof??
HallsofIvy said:How are you defining "e"?
HallsofIvy said:That's a perfectly good proof Eidos- provided you have already proved that
[tex]\frac{de^x}{dx}= e^x[/tex]
without using that limit. And you can do that if you start from the right definition of e.
Specifying a single value removes the ambiguity.f(x)= ex is defined as the function, y, satisfying the differential equation dy/dx= y, together with the initial value y(0)= 1.
Eidos said:Could you use this as the definition for e?
[tex]\frac{de^x}{dx}= e^x[/tex]
sushrutphy said:hey guys...there's another value for lim (1+x)^1/x..it goes like this...
e(1 - x/2 + 11x^2/24 ...)
found it in one of the books of higher math...but i can't find its proof...can anyone help me out?
The expression lim (x+1)^(1/x) represents the limit of the function (x+1)^(1/x) as x approaches infinity. In other words, it represents the value that the function approaches as x gets larger and larger.
To prove that lim (x+1)^(1/x)=e, we use the definition of a limit and apply algebraic manipulation and the properties of exponents. We can also use the fact that e is the limit of (1+1/n)^n as n approaches infinity, and rewrite the original expression as (1+1/x)^x. Then, we can use substitution and the definition of e to show that the limit is equal to e.
The number e is a mathematical constant that is approximately equal to 2.71828. It appears in many areas of mathematics and science, such as in exponential and logarithmic functions, compound interest, and population growth. Therefore, proving that lim (x+1)^(1/x)=e holds a lot of significance and shows the connection between these different concepts.
No, this limit can only be evaluated for values of x that are approaching infinity. This is because the expression (x+1)^(1/x) becomes undefined when x is equal to 0 or negative values, and the limit only applies to values of x that are getting larger and larger.
This limit is used in many practical applications, such as in finance, biology, and physics. For example, it can be used to calculate compound interest or to model population growth. It also plays a role in optimization problems and in analyzing the behavior of functions as their inputs approach infinity.