Solving Limit of (1-cos(2x^2)) / (1-cos(3x^2)): Is There Another Way?

  • Thread starter kira137
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Instead, it gives:\lim_{x\rightarrow 0}\frac {4\cos 2x^2 - 8x^2\sin 2x^2} {6\cos 3x^2 - 18x^2\sin 3x^2}To solve this, you can use the trigonometric identity:\sin 2x = 2\sin x\cos xAnd then use the fact that:\lim_{x\rightarrow 0} x^n\sin x = 0In summary, the conversation discusses finding the limit of a function involving trigonometric expressions as x approaches 0. The initial attempt at solving using l'Hopital's rule leads
  • #1
kira137
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Homework Statement


Find

limit ...(1-cos(2x^2)) / (1-cos(3x^2))
x->0

2. The attempt at a solution
since the above gave me 0/0, I used l'Hopital method..
then i got

(4x)sin(2x^2) / (6x)(sin3x^2)
which gives me 0/0 again..

so I kept on using l'Hopital.. but it seemed on going forever

is there other way to solve this..?
thank you in advance
 
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  • #2
kira137 said:

Homework Statement


Find

limit ...(1-cos(2x^2)) / (1-cos(3x^2))
x->0

2. The attempt at a solution
since the above gave me 0/0, I used l'Hopital method..
then i got

(4x)sin(2x^2) / (6x)(sin3x^2)
which gives me 0/0 again..

so I kept on using l'Hopital..

Don't keep using LH rule. Remember you know (at least you should know):

[tex]\lim_{x\rightarrow 0}\frac {\sin x} x = 1[/tex]

See if you can figure out how to use that next.
 
  • #3
Using LH twice doesn't give 0/0.
 

Related to Solving Limit of (1-cos(2x^2)) / (1-cos(3x^2)): Is There Another Way?

1. What is the limit of (1-cos(2x^2)) / (1-cos(3x^2)) as x approaches 0?

The limit of this expression as x approaches 0 is 1.

2. Why is the limit of (1-cos(2x^2)) / (1-cos(3x^2)) equal to 1?

This is because both the numerator and denominator approach 0 as x approaches 0, and the limit of their quotient is equal to the limit of their individual parts multiplied together.

3. Is there another way to solve this limit?

Yes, there is another way to solve this limit using L'Hopital's rule, which states that the limit of a quotient of two functions is equal to the limit of the derivatives of the numerator and denominator.

4. Can this limit be solved without using calculus?

Yes, this limit can also be solved using trigonometric identities and algebraic manipulations. However, this method may be more complex and time-consuming compared to using calculus.

5. What is the significance of solving this limit?

Finding the limit of a function helps us understand the behavior of the function at a specific point or as x approaches a certain value. In this case, solving the limit of (1-cos(2x^2)) / (1-cos(3x^2)) helps us understand the behavior of this expression at x=0, which can be applied to other similar problems in calculus and mathematics.

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