This is a summary of the concept of Limits and Continuity. This concept is crucial for learning advanced calculus, which includes topics such as differentiation and integration.
Limits and Continuity
Vague Definition of Limits
Let 𝑎 𝜖 ℝ and f be a function defined on an open interval containing 𝑎 (except possibly at 𝑎). Suppose L is a real number such that f(x) is as
close to L as we please whenever x is sufficiently close (but not equal to 𝑎). The number L is the limit of f(x) as x approaches 𝑎 and can be
written as lim 𝑓(𝑥) = 𝐿.
𝑥→𝑎
Limit Laws
Let a, c, L and M be real numbers and suppose that lim 𝑓(𝑥) = 𝐿 and lim 𝑔(𝑥) = 𝑀. Then
𝑥→𝑎 𝑥→𝑐
lim 𝑐𝑓(𝑥) = 𝑐 lim 𝑓(𝑥) lim [ 𝑓(𝑥) ± 𝑔(𝑥)] = lim 𝑓(𝑥) ± lim 𝑔(𝑥)
𝑥→𝑎 𝑥→𝑎 𝑥→𝑎 𝑥→𝑎 𝑥→𝑎
(Scalar Multiplicity) (Linearity)
lim [ 𝑓(𝑥) × 𝑔(𝑥)] = lim 𝑓(𝑥) × lim 𝑔(𝑥) lim 𝑓(𝑥)
𝑓(𝑥) 𝑥→𝑎
𝑥→𝑎 𝑥→𝑎 𝑥→𝑎
lim = provided that 𝑀 ≠ 0
𝑥→𝑎 𝑔(𝑥) lim 𝑔(𝑥)
𝑥→𝑎
One-sided Limits
If f(x) approaches L1 as x approaches a from the Left, we write: 𝑙𝑖𝑚−𝑓(𝑥) = 𝐿1
𝑥→𝑎
If f(x) approaches L1 as x approaches a from the Left, we write: 𝑙𝑖𝑚+𝑓(𝑥) = 𝐿2
𝑥→𝑎
Relationship between one-sided and two-sided limits
If both 𝑙𝑖𝑚− 𝑓(𝑥) and 𝑙𝑖𝑚+ 𝑓(𝑥) exist and 𝑙𝑖𝑚−𝑓(𝑥) = 𝑙𝑖𝑚+ 𝑓(𝑥), then 𝑙𝑖𝑚𝑓(𝑥) exists and 𝑙𝑖𝑚𝑓(𝑥) = 𝑙𝑖𝑚− 𝑓(𝑥) = 𝑙𝑖𝑚+ 𝑓(𝑥).
𝑥→𝑎 𝑥→𝑎 𝑥→𝑎 𝑥→𝑎 𝑥→𝑎 𝑥→𝑎 𝑥→𝑎 𝑥→𝑎
The converse holds true as well.
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