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

• kira137
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
kira137

## 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..?

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):

$$\lim_{x\rightarrow 0}\frac {\sin x} x = 1$$

See if you can figure out how to use that next.

Using LH twice doesn't give 0/0.

## 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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