Question

Find \(lt_{(x,y)\rightarrow(0,1)}\frac{x+y-1}{\sqrt{x+y}-1}\)

(a) 9

(b) 0

(c) 6

(d) 2

This question was posed to me in final exam.

The origin of the question is Limits and Derivatives of Several Variables in division Partial Differentiation of Engineering Mathematics

Answer 1

Correct answer is (d) 2

To explain: Simplifying the expression we have

=\(lt_{(x,y)\rightarrow(0,1)}\frac{(\sqrt{x+y}^2)-(1)^2}{\sqrt{x+y}-1}=lt_{(x,y)\rightarrow(0,1)}\frac{(\sqrt{x+y}+1).(\sqrt{x+y}-1)}{\sqrt{x+y}-1}\)

=\(lt_{(x,y)\rightarrow(0,1)}(\sqrt{x+y}+1)=\sqrt{1}+1\)

=2