Let S be a subset of [itex]R[/itex], let a be a real number or symbol [itex]+ \inf[/itex] or [itex]- \inf[/itex] that is the limit of some sequence in S, and let L be a real number or symbol [itex]\inf[/itex] or [itex]- \inf[/itex]. We write [itex]lim_{x \rightarrow a^s} f(x) =L[/itex] if(adsbygoogle = window.adsbygoogle || []).push({});

1) f is a function defined on S,

2) for every sequence [itex](x_n)[/itex] in S with limit a, we have [itex]lim_{n \rightarrow \inf} f(x_n) = L[/itex]

a) For [itex]a \in R[/itex] and a funciton f we write [itex]lim_{x \rightarrow a} f(x) =L[/itex] provided [itex]lim_{x \rightarrow a^s} f(x) = L[/itex] for some set S = J \ {a} where J is an open interval containing a. [itex]lim_{x \rightarrow a} f(x) =L[/itex] is called the [two-sided] limit of f at a. Note that f need not be defined at a and, even if f is defined at a, the value f(a) need not equal [itex]lim_{x \rightarrow a} f(x) = L[/itex]. In fact, [itex]f(a) = lim_{x \rightarrow a} f(x) [/itex] if and only if f is defined on an open interval containing a and f is continuous at a.

Can anybody please explain this to me...it just seems very confusing...

a) Why can't {a} be contianed in S? (When it says "[itex]lim_{x \rightarrow a^s} f(x) = L[/itex] for some set S = J \ {a} where J is an open interval containing a")?

b) Also, this seems kind of strange to me...

First it says "Note that f need not be defined at a and, even if f is defined at a, the value f(a) need not equal [itex]lim_{x \rightarrow a} f(x) = L[/itex]"...and then it says "[itex]f(a) = lim_{x \rightarrow a} f(x) [/itex] if and only if f is defined on an open interval containing a and f is continuous at a.".

So if f is defined on an open interval containg a, then it must also be defined at a, right? Then why does it say that f doesnt need to be defiend at a? Also, if f is continuous at a ,shouldn't it be defined there too? Finally, why does the value f(a) not need to equal [itex]lim_{x \rightarrow a} f(x) = L[/itex]?

Can anybody please help me understand this?

Thanks in advance.

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# Limits of functions

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