Approximate Formula for Large x: Understand Derivation

In summary, the approximate formula for large x is given by f(x) ≈ exp(x), where exp(x) is the exponential function. This formula is derived using the Taylor series expansion of the exponential function and keeping only the first term, which is x. Its significance lies in its ability to estimate the value of the exponential function for large values of x, where the higher order terms become negligible. However, this formula is only accurate for large values of x and may not hold true for smaller values where the higher order terms become important. Moreover, this formula is specific to the exponential function and cannot be applied to other functions.
  • #1
Yegor
147
1
There exist such a formula for large x:
[tex]\arctan(x)\approx \pi/2-1/x+1/3x^3...[/tex]
I can't understand how it is derived. I tried to get it from Taylor series (for x -> infinity) and understood that here is something different. Can someone help me?
Thank you
 
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  • #2
have you tried arctan(1/x) for x->0?

and note:
arctan(1/x) = Pi/2 - arctan(x)
 
Last edited:
  • #3
Hm. This is what i got:
[tex] f(x)=\arctan(1/x);
f'(x)=-\frac{1}{1+x^2};
f(x)\approx f(0)+f'(0)(x-0)=\pi/2-x[/tex]
yes. it looks good
 
  • #4
Great. i got next terms too. Thank you very much.
 

FAQ: Approximate Formula for Large x: Understand Derivation

What is the approximate formula for large x?

The approximate formula for large x is given by f(x) ≈ exp(x), where exp(x) is the exponential function.

How is the approximate formula derived?

The approximate formula is derived using the Taylor series expansion of the exponential function and keeping only the first term, which is x. This approximation works well for large values of x where the higher order terms become negligible.

What is the significance of large x in this formula?

The formula is specifically designed for large values of x, where the exponential function grows rapidly and becomes difficult to calculate. Using this approximate formula makes it easier to estimate the value of the exponential function for large values of x.

Are there any limitations to this approximate formula?

Yes, the formula is only accurate for large values of x. For smaller values of x, the higher order terms in the Taylor series become important and the approximation may not hold true. It is important to use the exact formula for smaller values of x.

Can this formula be used for other functions?

No, this formula is specifically derived for the exponential function and cannot be applied to other functions. Each function has its own unique Taylor series expansion and approximate formula for large values of x would need to be derived separately.

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