Question

The value of limx → 0⁡⁡ [x]Cos(x), [x] denotes the greatest integer function _______

(a) lies between 0 and 1

(b) lies between -1 and 0

(c) lies between 0 and 2

(d) lies between -2 and 0

This question was posed to me during a job interview.

My query is from Limits and Derivatives of Several Variables topic in portion Partial Differentiation of Engineering Mathematics

Answer 1

Correct answer is (b) lies between -1 and 0

Easiest explanation: limx → 0⁡⁡ [x]Cos(x)

We know that,

x-1 < [x] < x

Multiplying by Cos(x), we get

(x-1)Cos(x) < [x]Cos(x) < xCos(x)

Taking limits, we get

limx → 0 [(x-1)Cos(x)] < limx → 0 [x]Cos(x) < limx → 0[xCos(x)]

=> -1 < limx → 0 [x]Cos(x) < 0.