Calculus: Limits and Their Properties

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Intro to limits and their properties. (Symbols were not working when set created therefore old school/other means used) :)

Limit does not exist

f(x) approaches different number from right side of c than it approaches from left side of c

Limit does not exist

f(x) increases or decreases without bound as x approaches c

Limit does not exist

f(x) oscillates between two fixed values as x approaches c

Epsilon/Delta definition of a Limit

Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. Then: lim f(x) = L (as x-->c) means that for each Epsilon>0 there exists a Delta>0 such that if 0<[x - c]<Delta, then [f(x) - L]<Epsilon.

Derivative of a Function

f'(x) = lim(as x-->0) {f(x+x) - f(x)}/*x provided the limit exists. For all x for which this limit exists, f' is a function of x.

Intermediate Value Theorem (IVT)

If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c) = k

f is continuous if

f(c) is defined

f is continuous if

lim(as x-->c) f(x) exists

f is continuous if

lim(as x-->c) f(x) = f(c)

Squeeze Theorem

If h(x) <_f(x) <_ g(x) for all x in an open interval containing c, except possibly at c itself, and if lim(as x-->c) h(x) = L = lim(as x-->c) g(x) then lim(as x -->c) f(x) exists and is equal to L.

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