Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm having difficulty understanding how my calc teacher manipulated this problem

http://postimage.org/image/40or35jj5/

What I dont understand is how he minipulated the second line to the third line. In other words, I don't understand how he changed $6x^2$ to $6\cos3x$. And how he changed $\displaystyle \frac{1}{\sin2x}$ to $\displaystyle \frac{x}{\sin2x}$. Unfortunately I have no way of getting in contact with him and the test is tomorrow. Any help would be greatly appreciated.

sorry about the external URL, I just don't know of any quicker way to add the math to this post.

share|improve this question

3 Answers 3

up vote 1 down vote accepted

He didn't change anything. Note that $x^{2}=x \cdot x$ so that by changing $6x^{2} \cos 3x$ to $6 \cos 3x$, that "frees up" the $x^{2}$ to make the other changes such as changing $\frac{1}{\sin 2x}$ to $\frac{x}{\sin 2x}$.

share|improve this answer

That's not what your teacher did. He rewrote the entire expression:

\begin{align} 6x^2 \cot 3x \csc 2x &= 6x^2\cdot \frac{\cos 3x}{\sin 3x} \cdot \frac{1}{\sin 2x}\\ &= 6\cos 3x \cdot \frac{x}{\sin 3x} \cdot \frac{x}{\sin 2x} \end{align}

and then he took the limit of each factor separately.

share|improve this answer

Hint:

Utilize following two identities :

$$\lim_{y\to0}\frac{\sin y}y=1\text{ and } \lim_{y\to0}\cos y=1$$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.