If, lim [(A)/2^y] as y->infinity = (1/2)*[(1/2)^x], what is A(x)?

  • Thread starter redphoton
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In summary, the notation "lim [(A)/2^y] as y->infinity = (1/2)*[(1/2)^x]" represents the limit of the function [(A)/2^y] as y approaches infinity, which is equal to half of the function [(1/2)^x]. A(x) represents the value of the function at a specific value of x and can be solved for by using algebraic manipulation. There are restrictions on the values of x and A in this equation, and it can be applied to any function that satisfies the given limit, particularly exponential functions.
  • #1
redphoton
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If

lim [A(x,y)/2^y] as y->infinity = (1/2)*[(1/2)^x] ,

What is A(x,y)?
 
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  • #2
You would want to have a term that cancels out the denominator. For example take A(x,y)=C(x)D(y), if we then take D(y)=2^y [itex]\lim_{y \to \infty}A(x,y)/2^y=\lim_{y \to \infty}C(x)=C(x)[/itex].
 
  • #3
A(x,y) must not be y independent for y<infinity.
 

1. What is the meaning of the notation "lim [(A)/2^y] as y->infinity = (1/2)*[(1/2)^x]"?

The notation "lim [(A)/2^y] as y->infinity = (1/2)*[(1/2)^x]" represents the limit of the function [(A)/2^y] as y approaches infinity, which is equal to half of the function [(1/2)^x]. This means that as y gets closer and closer to infinity, the function [(A)/2^y] approaches half of the function [(1/2)^x].

2. What is the significance of A(x) in this equation?

A(x) represents the value of the function at a specific value of x. In this equation, we are trying to find the value of A(x) that satisfies the given limit as y approaches infinity.

3. How can I solve for A(x) in this equation?

To solve for A(x), we can use algebraic manipulation to isolate A(x) on one side of the equation. First, we can multiply both sides of the equation by 2^y to get A = 2^y * [(1/2)^x]. Then, we can take the logarithm of both sides to get log(A) = y * log(2) + x * log(1/2). Finally, we can solve for A by raising both sides to the power of e, giving us A = e^(y * log(2) + x * log(1/2)).

4. Are there any restrictions on the values of x and A in this equation?

Yes, there are some restrictions on the values of x and A in this equation. First, x cannot equal 0, as this would result in an undefined value for A(x). Additionally, A must be a positive number, since we cannot take the logarithm of a negative number.

5. Can this equation be applied to any type of function?

Yes, this equation can be applied to any function that satisfies the given limit. However, the values of A and x may vary depending on the specific function being used. Additionally, this equation is typically used with exponential functions, as shown in the given example.

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