By Bowman J.C.
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Extra resources for Math 117-118, Honours Calculus
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Notice in this case that lim f (x) = 0 = f (0). However, this is not true for all functions f . x→0 The value of a limit as x → a might be quite different from the value of the function at x = a. Sometimes the point a might not even be in the domain of the function, but the limit may still be defined. This is why we restrict 0 < |x − a| (that is, x = a) in the above definition. Remark: The value of f at a itself is irrelevant to the limit. We don’t need to evaluate f at x = a any more than we need to evaluate 1/n at n = ∞ to find its limit.
12 (Limit Superior and Limit Inferior): Let {an }∞ n=1 be a bounded se∞ quence. Consider the sequence {sn }n=1 defined by sn = sup{an , an+1 , an+2 , . } for n ∈ N. (a) Prove that {sn }∞ n=1 is a bounded sequence. ∞ (b) Prove that {sn }∞ n=1 is a monotone sequence. Is {sn }n=1 an increasing or a decreasing sequence? (c) Prove that {sn }∞ n=1 is convergent. Note: The limit of the sequence {sn }∞ n=1 is known as the limit superior of the ∞ sequence {an }n=1 and is written lim sup an . This is just the supremum of the valn→∞ ues in the tail of the sequence.
An } always has a maximum element max(a1 , a2 , . . , an ). Note that max(a1 , a2 , . . , an ) ≥ ai for i = 1, 2, . . , n. • Note that [0, 1] has maximum element 1, but [0, 1) has no maximum element. Definition: Let S ⊂ R. Suppose there exists a real number a such that (i) x ≥ a for each x ∈ S (a is a lower bound for S), (ii) If ℓ is a lower bound of S, then a ≥ ℓ. Then a is called the greatest lower bound, or infimum, of S. b. S or a = inf S. Definition: If a = inf S and a ∈ S, we say a is the minimum of S.