Expert: Paul Klarreich - 1/27/2006

Question
In the definition of the derivative as the limit of the difference quotient we say that the limit, if exists, must be finite. Why the limit must be finite?

VT

Answer
Hi, Viktor,

[THERE WAS SOMETHING WRONG AT THE ALLEXPERTS SITE, SO THERE WAS SOME DELAY IN SENDING THIS.]

Your Question:  In the definition of the derivative as the limit of the difference quotient we say that the limit, if exists, must be finite. Why must the limit be finite?
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Because the limit must be a number, and infinity is not the name of a number.  You have, I assume, learned the formal definition of a limit, i.e. that:

Notation:
1) lim(x->c)f(x)  sounds like:  "the limit as x approaches c of f(x)"]
2) I can't make epsilons and deltas, so I use e and d.


lim(x->c)f(x) = L, if for any e>0, there exists d>0 such that whenever |x-c|<d, |f(x)-L|<e

That requires L to be an actual number.  But if we say, for example, that

lim(x->c) f(x) = +inf

There is no way we can write  |f(x)-inf| < anything, because infinity isn't a number.  So the infinity in that expression is not a value of L.  The meaning of that sentence is:

lim(x->c)f(x) = +inf, if for any M>0, there exists d>0 such that whenever |x-c|<d, f(x) > M

which is a different kind of statement from |f(x)-L| < e.