How to Find the Limit of sin(3x²)/(8x) as x Approaches Zero?

[Dan350]

1.
what would be the limit?? without using the L'Hopital's rule

lim_(x-0) (sin(3 x^2))/(8 x)


the limit of sin(3x^2) divided by 8x as x approaches zero











2. Limits of trignometric functions



3. The Attempt at a Solution
I tried factoring out the 1/8, but that's it, Idk how to go on


Hope you can help me

Thanks

 

[Dick]

1.
what would be the limit?? without using the L'Hopital's rule

lim_(x-0) (sin(3 x^2))/(8 x)the limit of sin(3x^2) divided by 8x as x approaches zero


2. Limits of trignometric functions
3. The Attempt at a Solution
I tried factoring out the 1/8, but that's it, Idk how to go onHope you can help me

Thanks

What's lim x->0 of sin(3x^2)/(3x^2)? That should be a good hint.

 

[Dan350]

What's lim x->0 of sin(3x^2)/(3x^2)? That should be a good hint.

1


but how to put the 3x^2 below?

 

[Curious3141]

1but how to put the 3x^2 below?

What (expression) do you have to multiply 8x by to get 3x²? As long as you multiply both the top and bottom by the same, you don't change the fraction.

 

[Dan350]

What (expression) do you have to multiply 8x by to get 3x²? As long as you multiply both the top and bottom by the same, you don't change the fraction.

3x on both numerator and denominator?
i would get 24x^2, but i can split the fraction right?

 

[Curious3141]

3x on both numerator and denominator?
i would get 24x^2, but i can split the fraction right?

So you get $\frac{3x\sin(3x^2)}{8(3x^2)}$. What happens next?

 

[Dan350]

So you get $\frac{3x\sin(3x^2)}{8(3x^2)}$. What happens next?

We factor out.

then the limx-0 of 3x/8 and limx-0 of sin(3x^2)/3x^2
therefore

limx-0 0*1
so

limx-0f(x)= 0

 

[Curious3141]

We factor out.

then the limx-0 of 3x/8 and limx-0 of sin(3x^2)/3x^2
therefore

limx-0 0*1
so

limx-0f(x)= 0

Yes, exactly.