- #1
Alpharup
- 225
- 17
he
definition of the limit of a function is as follows:[5]
Let
be a function defined on a subset [PLAIN]https://upload.wikimedia.org/math/a/1/b/a1b67abab803e714098f3e69a33900da.png, let
be a limit point of [PLAIN]https://upload.wikimedia.org/math/f/6/2/f623e75af30e62bbd73d6df5b50bb7b5.png, 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, 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, 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". 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 existence of such an h? Also, for what all h does this inequality o<|h|<δ implies |f(c+h)-L|<ε hold?
Let
the function
is defined to mean
for all [PLAIN]https://upload.wikimedia.org/math/b/0/f/b0f19c5714fe9f9891ed26ff783cf639.png, there exists a https://upload.wikimedia.org/math/1/c/b/1cb24dafc8035d2c720256620066ae73.png such that for all
.
Taken from https://en.wikipedia.org/wiki/(ε,_δ)-definition_of_limit
The problem what Iam facing is the statement ," for all
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 existence of such an h? Also, for what all h does this inequality o<|h|<δ implies |f(c+h)-L|<ε hold?
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