Does limit exist as x approaches zero? Frobenius Method DEQ

In summary, the conversation is about finding the limit of a function (4x^2-1)/(4x^2) as x approaches 0, and whether L'Hopital's Rule can be used to solve it. The conversation includes a question about using L'Hopital's Rule twice and confusion about the use of the Frobenius Method. Ultimately, it is determined that the limit is negative infinity.
  • #1
lonewolf219
186
2

Homework Statement



what is the limit of (4x^2-1)/(4x^2)
when x→0

Homework Equations



In order to find the Indicial Equation, do I need to take the limit of p(x) and q(x), the non-constant coefficients? If so, can the limit of this function be found using LH Rule?

The Attempt at a Solution



Please let me know any info you might have about the Frobenius Method, since I am just learning it from my professor's brief notes about it...
 
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  • #2
lonewolf219 said:

Homework Statement



what is the limit of (4x^2-1)/(4x^2)
when x→0


What do you think? What have you tried?
 
  • #3
Well, can you use L' Hopital's Rule twice? (8x - 0)/(8x) and then (8/8) = 1 ?

But I'm confused if I need to multiple by x^2 to find q(nault)?

x^2*q(x)=q(nault)

y''(x) + p(x)y'(x) + q(x)y(x) = 0

If so, would the limit be -1/4?

x^2(4x^2-1)/(4x^2) = [(4x^4)/4x^2] - [x^2/(4x^2)] = [x^2] - [1/4] = [x=0] = - 1/4
 
Last edited:
  • #4
L'Hopital applies only when the numerator and denominator both go to 0. Here, if x= 0, the numerator is -1 but the denominator is 0. Suppose x were some very small number, say x= 0.000001. What would that fraction be? Now, what do you thing the limit is?
 
  • #5
negative infinity, right?
 

FAQ: Does limit exist as x approaches zero? Frobenius Method DEQ

1. What is the definition of a limit?

A limit is a mathematical concept that describes the behavior of a function as its input approaches a certain value. It represents the value that the function approaches or “approaches closer and closer to” as the input gets closer and closer to a specific value.

2. How do you determine if a limit exists?

A limit exists if the function approaches the same value regardless of the direction from which the input approaches the specified value. This is known as the two-sided limit. If the one-sided limit from the left and the one-sided limit from the right are equal, then the limit exists.

3. What is the Frobenius method for solving differential equations?

The Frobenius method is a technique for solving differential equations with a regular singular point. It involves assuming a power series solution and using the recurrence relation between the coefficients of the series to find the solution.

4. How is the limit as x approaches zero related to the Frobenius method for solving DEQs?

The limit as x approaches zero is used in the Frobenius method to determine the convergence of the power series solution. If the limit is non-zero, the series converges and can be used to find a solution to the differential equation. If the limit is equal to zero, the series may still converge, but further analysis is needed.

5. Can the Frobenius method be used for any type of differential equation?

No, the Frobenius method can only be used for differential equations with a regular singular point. This means that the equation must have a term in the form of (x-a)^n, where a is the singular point and n is a non-negative integer. If the equation does not have this form, a different method must be used to solve it.

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