ABSTRACT
We saw in a previous exercise that limθ→0 cos θ = 1, and so application of the squeeze theorem for functions to the double inequality (4.8) yields the result. "
Our final order property results pertaining to limits of real-valued functions have to do with one-sided limits. In particular, for f : (a, b) → R and x0 ∈ [a, b), we wish to investigate the behavior of f (x) as x → x0 from the right, i.e., through values of x > x0. Similarly, for x0 ∈ (a, b], we wish to investigate the behavior of f (x) as x→ x0 from the left, i.e., through values of x < x0. A careful look at Definition 4.1 on page 156will show that such one-sided limits are already well defined, since that definition handles the case where x0 is a limit point. Of course, this includes limits such as limx→a f (x) or limx→b f (x) for a function f : (a, b)→ R. However, we make this idea more explicit in the case of real-valued functions through the following special definition.
