Lim x-0 ([x/sinx] -[sinx/x])/[tanx/x] where [] represents greatest integer function. what is the value of this limit? ans:1?

Question

Lim x->0 ([x/sinx] -[sinx/x])/[tanx/x] where [] represents greatest integer function. what is the value of this limit? ans:1?

Answer

Solution:

Let’s take [x/sinx] as n. Then,

lim x->0 (n - [sinx/x])/[tanx/x]

Let's first calculate n. For n to be defined,

sinx = x

x^2 = sin^2x

x^2/sin^2x = 1

As x approaches 0, x/sin x approaches 1, and sin x/x approaches 1.

Therefore, n = [1] = 1.

Now, let's calculate the numerator:

n - [sinx/x] = 1 - 1 = 0.

Now, let's calculate the denominator:

tan x = sin x/cos x = (sin x/x)/(cos x/x)

As x approaches 0, cos x/x approaches 1. Therefore,

tan x/x = sin x/x.

Therefore, the limit becomes:

lim x->0 0/sin x/x = 0/1 = 0.

Therefore, the value of the limit is 0.