Question

Find \(=lt_{x\rightarrow 0}\frac{(a_nx^n+a_{n-1}x^{n-1}+…+a_1x+a_0)}{(b_nx^n+b_{n-1}x^{n-1}+…+b_1x+b_0)}\)

(a) ^an ⁄ bn

(b) ∞

(c) No general form

(d) ^bn ⁄ an

This question was addressed to me in semester exam.

This is a very interesting question from Indeterminate Forms topic in section Differential Calculus of Engineering Mathematics

Answer 1

The correct choice is (a) ^an ⁄ bn

Easiest explanation: \(=lt_{x\rightarrow 0}(\frac{x^n}{x^n})\times\frac{(a_n+a_{n-1}\frac{1}{x}+…+a_1\frac{1}{x^{n-1}}+a_0\frac{1}{x^{n}})}{(b_n+b_{n-1}\frac{1}{x}+…+b_1\frac{1}{x^{n-1}}+b_0\frac{1}{x^{n}})}\)

\(=\frac{a_n}{b_n}\)