Find $\lim _{x \to 0} \cot(3x)\sin(4x)$

I need to find $$\lim _{x \to 0} \cot(3x)\sin(4x)$$. However, I am having trouble finding a way to do that. I am a Calculus 1 student and the only ways I know to handle a problem like this are by multiplying by a conjugate, or L'Hospital's Rule. Neither of which seems to work here.

I think I need to identify the correct trig identity for cotangent or sine, and then apply one of the two methods I mentioned, but I can't seem to find any trig identities that seem to allow that.

I have tried replacing $$\cot(3x)\sin(4x)$$ with $$\frac{\cot(3x)}{\csc(4x)}$$ and $$\frac{\frac{cos(3x)}{sin(3x)}}{\csc(4x)}$$ and $$\frac{\cos(3x)\sin(4x)}{\sin(3x)}$$, but I can't find any that work.

As I understand it, L'Hospital's rule requires that both the numerator and denominator of the limit approach either zero or infinity. I'm not entirely sure what that means, but $$(0, 0)$$ is a point on both the graph of $$\cos(3x)\sin(4x)$$ and $$\sin(3x)$$.

I can do,

$$\lim _{x \to 0}\frac{\frac{d}{dx}\cos(3x)\sin(4x)}{\frac{d}{dx}\sin(3x)}$$

$$=\lim_{x \to 0} \frac{[-3\sin(3x)\cdot\sin(4x)] + [\cos(3x)\cdot4\cos(4x)]}{\cos(3x)}$$

$$=\lim_{x \to 0} \frac{4\cos(3x)\cos(4x)-3\sin(3x)\sin(4x)}{\cos(3x)}$$

$$=\frac{4(1)(1) - 3(0)(0)}{1} = 4$$

But this is wrong.

How can I solve this limit?

• You are nearly there, $\frac{d}{dx}\sin{(3x)} = \color{red}3\cos{(3x)}$. The other answers below are the usual right way to do it though. – user1952500 Apr 7 at 1:04
• @user1952500 That was all I needed, thank you! – LuminousNutria Apr 7 at 1:06

2 Answers

Hint:$$cot(3x)sin(4x)=cos(3x){x\over {sin(3x)}}{{sin(4x)}\over x}$$

Use the fact that $$lim_{x\rightarrow 0}{{sin(x)}\over x}=1$$.

$$\cot(3x)\sim_0 \frac{1}{3x}$$ and $$\sin{4x}\sim_0 4x$$ $$...$$ $$\cot{3x}\sin{4x}\sim_0 \frac{4}{3}$$