I am given that $$\lim_{x\rightarrow 0}\sin(mx)\cot\left(\frac{x}{\sqrt{3}}\right) = 2$$ and invited to solve for $m$.
My approach is like this. Re arrange the limit like so:
$$
\lim_{x\rightarrow 0}{\sin(mx) \cos(\frac{x}{\sqrt{3}})} = 2\times \lim_{x\rightarrow 0}{\sin(\frac{x}{\sqrt{3}})}
\\
\lim_{x\rightarrow 0}{\sin(mx) \cos(\frac{x}{\sqrt{3}})} = 0
\\
\lim_{x\rightarrow 0}{\sin(mx)} = 0
$$
My conclusion is that the value of $m$ is immaterial: the limit will go to zero anyway.
Where am I going wrong?
Best Answer
What you've shown is that $\lim_{x\to0} \sin(mx)=0$ for every $m$, which is true. However, this tells you nothing about the limit $$\lim_{x\to0} \sin(mx)\cot(x/\sqrt 3),$$ which is an indeterminate form $0\cdot\infty$. You still have the job of determining what $m$ needs to be in order to resolve the indeterminate form into the value $2$. To do that, you should ignore the future result $2$ and simply work out the indeterminate form: $$ \lim_{x\to0} \sin(mx)\cot(x/\sqrt 3)=\lim_{x\to0}\frac{\sin(mx)\cos (x/\sqrt 3)}{\sin(x/\sqrt 3)}\tag1 $$ The RHS of (1) now has the form $\frac 00$, which you can evaluate with L'Hopital's rule. The answer will depend on $m$.