Trigonmetric limits

[kjbohn]

Find the limit as x approches 0. \(\displaystyle \lim_{x\to 0}\frac{(sin^{2}(4x^{2})+tan(2x^{4})+sin(x^{3})tan(3x))}{(x^{4}+x^{3}sin(2x)+x^{2}sin(x)tan(x)+xsin^{2}(x)tan(3x))}\).
So far I have turned the tangents in to sinx/cosx and that is it. I was wondering if you could then combine them together through multiplying them to get common denominatiors. Otherwise I am at a loss for words. Of you could please help.
Thank you.

 

[Deleted member 4993]

Find the limit as x approches 0. \(\displaystyle \lim_{x\to 0}\frac{(sin^{2}(4x^{2})+tan(2x^{4})+sin(x^{3})tan(3x))}{(x^{4}+x^{3}sin(2x)+x^{2}sin(x)tan(x)+xsin^{2}(x)tan(3x))}\).
So far I have turned the tangents in to sinx/cosx and that is it. I was wondering if you could then combine them together through multiplying them to get common denominatiors. Otherwise I am at a loss for words. Of you could please help.
Thank you.

Have you learned L'Hospital's Theorem?

 

[kjbohn]

No, not yet.:/ He said we were not allowed to use that yet.

 

[galactus]

This thing appears to me rather monstrous. Even with L'Hopital, the derivatives would be onerous.

Perhaps it is OK if you just use values getting closer and closer to 0.

Try 1, .1, .01. .001, and so on and see what it converges to.