MATH 1200 Midterm: Review for Midterm 2Premium
3 pages85 viewsFall 2016
SchoolUniversity of Guelph
Course CodeMATH 1200
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November 19, 2016
Dr. Matt Demers
Calculus midterm 2
Formal deﬁnition of a limit
For any limit limx→af(x) = L, we can deﬁne:
∀ϵ > 0∃δ > 0, s.t. if 0<|x−a|< δ, then |f(x)−L|< ϵ
Limits to inﬁnity
We can construct an argument similar to δ−ϵ, for limits to inﬁnity. The argument is as follows:
x→af(x) = ∞
∀N > 0∃δ > 0, s.t. if 0<|x−a|< δ, then f(x)> N.
where you start with f(x)> N, and work towards 0<|x−a|< δ.
Note, if the limit is −∞, then use N < 0.
x→∞ f(x) = L
∀ϵ > 0∃M > 0s.t. if x > M, then |f(x)−L|< ϵ
where you start with f|(x)−L|< ϵ, and work towards M > 0.
Note, if approaching −∞, then use x < M and M < 0.
x→∞ f(x) = ∞
∀N > 0∃M > 0, s.t. if x > M, then f(x)> N.
Note, if approaching −∞, then use x < M and M < 0— if the limit is −∞, then use N < 0.
Intermediate value theorem
If f(x)is continuous on [a, b], then for any y on that interval value, there must be x value that satisﬁes it.
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