Given that 1 - x2/6 smaller of same as x sin x / 2(1-cos x) smaller or same as 1. Use the Sandwich Theorem to obtain the limit of x sin x / 1 - cos x when x is 0

To find the limit of the function x sin(x) / (1 - cos(x)) as x approaches 0, we can use the Sandwich Theorem (also known as the Squeeze Theorem). The idea behind this theorem is to find two other functions that "squeeze" the given function between them and have the same limit at the point of interest.

Let's solve this step by step:

1. First, let's simplify the given inequality:
1 - x^2/6 ≤ x sin(x) / (2(1 - cos(x))) ≤ 1

2. Next, we can find the limit of each of the functions above as x approaches 0. Let's start with the left function:
lim(x→0) (1 - x^2/6) = 1 (since x^2/6 approaches 0 as x approaches 0)

Now, let's find the limit of the right function:
lim(x→0) 1 = 1 (this one is straightforward)

So, the limits of the left and right functions as x approaches 0 are both 1.

3. Now, we need to find a function that lies in between the two functions above. Consider x itself:
lim(x→0) x = 0

Since x approaches 0, we can use x as the middle function for our sandwich.

4. Finally, we conclude that using the Sandwich Theorem, the limit of x sin(x) / (1 - cos(x)) as x approaches 0 is also 1.