Suppose lim f(x)=4 as x approaches 5?

Question

Consider the following statements:
A) As x approaches 5, f(x) approaches 4
B) f is defined on (a, 5) u (5, b) for some a<5 C) Range of f contains 4

I dont get b but would a and b be right?

Answer 1

A is the only correct answer.

B: The statement does guarantee that there is no break between a and 5, and, 5 and b, meaning that f may not always be defined.

C: The range of f does not have to contain 4. As long as when x approaches 5 from both sides (left and right), the limit exists.

Answer 2

A is correct, being just a restatement of [x→5]lim f(x)=4.

B is actually false, but a lot of textbooks write this as one of the requirements for a limit to exist at c. However, this requirement is overly restrictive -- as long as c is an accumulation point of the domain of the function, the limit at c can be sensibly defined. Still, if your textbook is one of those that uses the more restrictive requirement, your teacher may expect you to answer this one as true. Check your textbook.

As an aside, the reason for requiring c to be an accumulation point is that, if it were not, one could show that any real number is the limit of f(x) as x→c, L by choosing for every ε>0 a δ>0 such that no points in (c-δ, c+δ)\{c} are actually in the domain of f -- thereby making it vacuously true that 0<|x-c|<δ → |f(x)-L|<ε, simply because there are no x in the domain of f such that 0<|x-c|<δ. Thus, we have to require that for every δ>0 that there actually are some points x in the domain of f such that 0<|x-c|<δ -- that is, that c is an accumulation point of the domain of f.

C is just plain false. There is no requirement in any definition that f ever actually attains the value 4.

Answer 3

a would be right.

If I understand your notation, than b should be right. f should be defined for some values less than 5 and greater than 5, otherwise how could you tell that there is a limit?

It has been a while since I thought about limits though - but my gut says that limits are only meaningful on conditions such that there is some continuity in the function as x approaches a certain value, even if f(x) {in this case, x=5} isn't defined.