Limits, one and two-sided

Hi Sir!

Sorry for this very elementary question. I want to ask about Analysis 1101
Topic: Limits of functions. What’s the difference between the these two
definitions?

1) Say lim f(x)= L as x→b‾. If given ε>0, we can find δ>0 such that when

b-δ< x <b =>│f(x)-L│< ε.

2) Assume f(x) is defined in an interval (a,b) with ξ ϵ (a,b) but possibly f is not defined at ξ. Say lim f(x)=L as x→ξ. If given ε >0, we can find δ>0 such that when 0 <│x-ξ│< δ =>│f(x)-L│< ε.

Thanks lots!
Smile always!
Vanessa =)

Oh! Is the difference in that (1) only accounts for the limit as x tends to a number from the negative side, so you will need to work out in a similar way to find the limit as it tends to the same number from the positive side?

Whereas (2) takes into account both limits as x tends to a number from both the negative and positive sides? But what happens when the graph experiences a drastic change when it goes from the negative side of a number to the positive side of the same number? Would the second definition still hold? If it doesn’t, would we have to split it into two parts like in (1)? And how would you know when the second definition holds or not?

An example of a graph with drastic change from lecture is:

f(x) = 4-x, x < 1
f(x)=4x, x ≥ 1.

Thanks and sorry to bother you on non-tutorial days…

Smile always!
Vanessa =)

Reply:

It seems you’ve already figured out the main point. I strongly advise all students to take note of this process: When the effort was expended to formulate the question in precise terms, the question essentially resolved itself! This is one of the main reasons I am recommending that people submit their questions in writing.

In fact, to define the meaning of lim_{x->c}f(x)=L, f never needs to be actually defined at c. Also, as you’ve figured out, the condition

0<|x-c|< delta

can be broken up as

c< x < c+ delta (x is quite close to the right of c)
*or*
c- delta < x < c (x is quite close to the left of c).

So the meaning of

lim_{x->c}f(x)=L

which is

`for all x close to c, f(x) is close to L,’

can be broken down into

`for all x close to c on the right *or* on the left, f(x) is close to L,’

On the other hand, the meaning of

lim_{x->c-}f(x)=L

is

`for all x close to the left of c, f(x) is close to L.’

So clearly, the first condition is stronger than the second.

Of course you can define the notion of right limits as well:

`lim_{x->c+}f(x)=L means …’

In fact, a minor theorem is:

lim_{x->c)f(x) exists if and only if
lim_{x->c-)f(x) and lim_{x->c+)f(x)

both exist *and* are equal to each other. In that case, of course,

lim_{x->c)f(x) =lim_{x->c-)f(x)=lim_{x->c+)f(x)

In the example you give above,

lim_{x->1+}f(x)=4 while lim_{x->1-}f(x)=3.

So lim_{x->1}f(x) does *not* exist.

If you take a function like

f(x)= x for x< 0, f(x)=1/x for x> 0,

then lim_{x->0-}f(x) exists, but {x->0+}f(x) does not. A fortiori, {x->0}f(x) does not exist.

If you define f(x)=1/x for all non-zero x, then neither the left nor right limit exists at 0. (They both exist and are equal at all other points, however.)

Note that in the preceding paragraphs, I’m being quite informal in describing the definitions using the phrase `close to.’ It’s important to acquire the ability to move freely between such casual descriptions that make meanings transparent, and the precise notions using epsilons and deltas.

MK

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