he definition of the limit of a function is as follows:[5](adsbygoogle = window.adsbygoogle || []).push({});

Let be a function defined on a subset [PLAIN]https://upload.wikimedia.org/math/a/1/b/a1b67abab803e714098f3e69a33900da.png, [Broken] let be a limit point of [PLAIN]https://upload.wikimedia.org/math/f/6/2/f623e75af30e62bbd73d6df5b50bb7b5.png, [Broken] and let be a real number. Then

the function has a limit at

is defined to mean

for all [PLAIN]https://upload.wikimedia.org/math/b/0/f/b0f19c5714fe9f9891ed26ff783cf639.png, [Broken] there exists a https://upload.wikimedia.org/math/1/c/b/1cb24dafc8035d2c720256620066ae73.png such that for all in that satisfy [PLAIN]https://upload.wikimedia.org/math/6/e/e/6ee6dc9ee03f04da6ff784b6eca416e5.png, [Broken] the inequality https://upload.wikimedia.org/math/2/c/4/2c412a6f49514db4dbd890537413bcc6.png holds.

.

Taken from https://en.wikipedia.org/wiki/(ε,_δ)-definition_of_limit

The problem what Iam facing is the statement ," for all in that satisfy [PLAIN]https://upload.wikimedia.org/math/6/e/e/6ee6dc9ee03f04da6ff784b6eca416e5.png". [Broken] Though I have understood this statement, this hinders my proving this property of limit

lim f(c+h)=L =lim f(x).

h->0 x->c

I am trying to prove the LHS from the RHS.

We know,

"the function f has a limit L at

is defined to mean

for every ε>0,there exists a δ>0, sucht that, for all x in D satisfying, o<|x-a|<δ, implies |f(x)-L|<ε.

The LHS can be proven from the trivial fact taking

x-c=h.

A question arises to me. How do we know the existance of such an h? Also, for what all h does this inequality o<|h|<δ implies |f(c+h)-L|<ε hold?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I Epsilon-Delta definition property.

Tags:

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**