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Keefer Rourke

November 19, 2016

MATH 1200

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:

case 1:

lim

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.

case 2:

lim

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.

case 3:

lim

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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