Define a neighborhood of – ∞ to be an open interval of the sort (– ∞, b), where b ∈ . Let c be a real number or ∞. (a) Use Exercise 16 to give reasonable definitions of limx → c f (x) = – ∞ and limx →...

Define a neighborhood of – ∞ to be an open interval of the sort (– ∞, b), where b ∈
. Let c be a real number or ∞.

(a) Use Exercise 16 to give reasonable definitions of limx → c f (x) = – ∞ and limx → – ∞ f (x) = c in terms of neighborhoods.

(b) Reformulate your answer to part (a) without referring to neighborhoods (similar to the style of Definitions 3.6 and 3.7).

(c) State and prove a sequential criterion for limits involving – ∞.

(d) State and prove results similar to l’Hospital’s rule for limits involving – ∞.

Exercise 16

Let x ∈ R. Recall that a neighborhood of x is an open interval of the sort (x – ε, x + ε), where ε > 0. We now define a neigh borhood of ∞ to be an open interval of the sort (a, ∞), where a ∈ R. Suppose that f : D → R and let c and L be real numbers or ∞. If c ∈ R, we require c to be an accumulation point of D, and if c = ∞, we require D to be unbounded above. Define limx → c f (x) = L if for every neighborhood V of L there exists a neighborhood U of c such that f (x) ∈ V for all x ∈ U ∩ D with x ≠ c. (It should also agree with Exercise 13 when c ∈ R and L = ∞.)