Mathematics

Problem

Two quotients of elementary functions are given below.

A:\displaystyle\; \dfrac{1-\cos x}{x^2}

x\to 0

B:\displaystyle\; \dfrac{\sin x- x}{x^3}

x\to 0

Answer the questions that follow. a. For each quotient

A

and

B

, decide whether the numerator and denominator both approach

0

x\to 0

. Support your conclusion with limit statements. b. For the quotient(s) in which both numerator and denominator approach

0

, discuss the additional differentiability requirements that must be verified on an open interval containing

x=0

(except possibly at

x=0

itself) before replacing the numerator and denominator by their first derivatives would be a logically sound next step in the analysis.