Right answer is (d) Does not Exist
To explain I would say: Multiplying and dividing by we have
\(lt_{(x,y)\rightarrow(\infty,0)}(sin(y))\times(\sum_{a=1}^{x-1}sin(\frac{a}{x}))\)
\(lt_{(x,y)\rightarrow(\infty,0)}(x.sin(y))\times lt_{(x,y)\rightarrow(\infty,0)}\left (\sum_{a=1}^{x-1}\frac{sin(\frac{a}{x})}{x}\right )\)
\(lt_{(x,y)\rightarrow(\infty,0)}(\frac{sin(y)}{\frac{1}{x}})\times lt_{(x,y)\rightarrow(\infty,0)}\left (\sum_{a=1}^{x-1}\frac{sin(\frac{a}{x})}{x}\right )\)
Put z=1/x : as x → ∞ : z → 0
Consider one part of the limit
\(=lt_{(x,y)\rightarrow (0,0)}\frac{sin(y)}{z}\)
Put : y = t : z = at
\(=lt_{t\rightarrow 0}\frac{sin(t)}{at}=\frac{1}{a} lt_{t\rightarrow 0}\frac{sin(t)}{t}\)
=\(\frac{1}{a}\times 1= \frac{1}{a}\).