GATE Overflow Test Series | Calculus | Test 1 | Question: 16

Question

If $\displaystyle \lim_{x \to 0}\sin{(mx)}\cot{\left (\frac{x}{\sqrt 3}\right)} = 2$ then $m = $ _________ (rounded off up to $2$ decimal places)

Answer 1

$\displaystyle \lim_{x \to 0}\sin{(mx)}\cot{\left(\frac{x}{\sqrt 3}\right)}$

 $\quad =\displaystyle \lim_{x \to 0}mx\frac{\sin{(mx)}\cot{\left(\frac{x}{\sqrt 3}\right)}}{mx}$
 
 Since, $\displaystyle \lim_{x \to 0 }\frac{\sin x}{x} = 1,$ given limit
 
 $\quad \displaystyle = \lim_{x \to 0}mx \cot{\frac{x}{\sqrt 3}}$
 
 $\quad \displaystyle = m. \lim_{x \to 0}x\dfrac{ \cos{\frac{x}{\sqrt 3}}}{\sin{\frac{x}{\sqrt 3}}}$
 
  $\quad \displaystyle = m.\sqrt 3 \lim_{x \to 0}\dfrac{\frac{x}{\sqrt 3}} {\sin{\frac{x}{\sqrt 3}}} \cos{\frac{x}{\sqrt 3}}$
 
 $\quad \displaystyle = m.\sqrt 3  \lim_{x\to 0} \cos{\frac{x}{\sqrt3}}. = m .\sqrt 3$
 
 Given that, $m.\sqrt 3 = 2 \implies m = \frac{2}{\sqrt3}=1.1547$