## anonymous 4 years ago 2nd question: Find lim (sin2xcot4x) X->0.

1. anonymous

are you allowed to use l'hopital's rule?

2. anonymous

assuming this is $\lim_{x\rightarrow 0}\sin(2x)\cot(4x)$ $\lim_{x\rightarrow 0}\frac{\sin(2x)\cos(4x)}{\sin(4x)}$

1/2

4. anonymous

in any case the answer is $\frac{1}{2}$

5. anonymous

but if you need to show your work your answer will depend on what you are allowed to use. l'hopital's rule is the simplest, otherwise it will be some work

6. anonymous

i dont know the hospital rule that you have said :(

7. anonymous

you mean you do not know it or you are not allowed to use it? i assume this is calc class, so hve you covered deriviatives yet or are you just starting out?

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without using l'hopital rule

10. anonymous

any computation as a long the answer will be the same in a long method

11. anonymous

i dont know it sorry :(

12. anonymous

there is a nice neat answer above. i think you can also use $\frac{sin(2x)\cos(4x)}{\sin(4x)}=\frac{\sin(2x)(\cos^2(2x)-\sin^2(2x))}{2\sin(2x)\cos(2x)}$

13. anonymous

here i used the double angle formula for sine and cosine

14. anonymous

can i post the 3rd question too?

15. anonymous

then cancel the sin(2x) top and bottom, get $\frac{\cos^2(2x)-\sin^2(2x)}{2\cos(2x)}$ replace x by zero, get $\frac{1}{2}$

16. anonymous

no limit to the amount you can post

17. anonymous

yes i will write your solution :)

18. anonymous

if f(x)=tanx-x and g(x)=x^3, evaluate the limit of f(x) over g(x) as x approaches 0. -3rd question.

19. anonymous

@nenadmatematica if you can do this without l'hopital or power series i will be impressed

haha I just wanted to ask you the same thing :D ....

21. anonymous

lol well, i guess we cannot give an elementary reason for this. the answer is $\frac{1}{3}$ but i cannot think of a gimmick to simplify this expression. are you sure you have not covered l'hopital? because i am stumped. in particular you have a trig fuction combined in combination with x and x^3 so there is no simple trig identity that will change the form of this for you

22. anonymous

@nenadmatematika tahnks for helping us too :)

well you're welcome....I agree with satellite that this example is very convenient for using L'Hopital rule.....I can't think of any other way now :D

24. anonymous

i really dont know but if both of you wants to use L'hopital rule. then i will agree to both of you

25. anonymous

4th question: Evaluate the lim x^2-16 over x+4 as x->4.