Question

Find \(lt_{(x,y)\rightarrow(\infty,0)}(\sum_{a=1}^{x-1}sin(\frac{a}{x}).sin(y))\)

(a) 1

(b) -1

(c) ∞

(d) Does not Exist

The question was posed to me during an online interview.

My question is from Limits and Derivatives of Several Variables topic in chapter Partial Differentiation of Engineering Mathematics

Answer 1

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

\(=lt_{t\rightarrow 0}\frac{sin(t)}{at}=\frac{1}{a} lt_{t\rightarrow 0}\frac{sin(t)}{t}\)

=\(\frac{1}{a}\times 1= \frac{1}{a}\).