Question

Find \(lt_{(x,y)\rightarrow (0,0)}\frac{y^7x^{98}-x^{97}y^8+x^{105}}{xy^7+x^8}\)

(a) Does Not Exist

(b) 0

(c) 1

(d) ∞

I had been asked this question in an international level competition.

The question is from Limits and Derivatives of Several Variables topic in division Partial Differentiation of Engineering Mathematics

Answer 1

The correct answer is (b) 0

The best I can explain: Put x =r.cos(ϴ) : y = r.sin(ϴ)

=\(lt_{(x,y)\rightarrow (0,0)}\frac{(r^7.sin^7(\theta))(r^{98}.sin^{98}(\theta))-(r^{97}.cos^{97}(\theta))(r^8.sin^8(\theta))+(r^{105}.cos^{105}(\theta))}{(r.cos(\theta)(r^7.sin^7(\theta))+(r^8.cos(\theta))}\)

=\(lt_{(x,y)\rightarrow (0,0)}\frac{r^{105}}{r^8}\times \frac{(sin^7(\theta))(sin^{98}(\theta))-(cos^{97}(\theta))(sin^8(\theta))+(cos^{105}(\theta))}{(cos(\theta)(sin^7(\theta))+(cos(\theta))}\)

=\(lt_{(x,y)\rightarrow (0,0)}(r^{97})\times \frac{(sin^7(\theta))(sin^{98}(\theta))-(cos^{97}(\theta))(sin^8(\theta))+(cos^{105}(\theta))}{(cos(\theta)(sin^7(\theta))+(cos(\theta))}\)

= 0