infinite limitation

Question

Suppose that f(x) is defined near x = −1 and satisfies −x^4 − x^2 + 4 􏰀≤ f(x) ≤ 􏰀 4 + 2x^2, calculate lim x→−1 f (x)

Answer 1

The limit is obviously 6...but I am hard pressed to explain exactly why.

This is an example of what sometimes is called the "sandwich" or "squash" principle, but that to me is usually explained poorly.

Clearly f(x) is always between the two given functions. Near -1 the function on the left is always a little less than 6 and the function on the right is always a little more than 6...and most importantly the 2 functions can be made as close to 6 as required, i.e.

given epsilon, there is a delta such that |f(x)-6|<epsilon whenever|x+1| <delta...and that is the definition of the limit!