Epsilon delta definition of limit.

• Oct 20th 2011, 10:56 AM
mervecetin
Epsilon delta definition of limit.
Q1) The precise definition of lim x->a (f(x)≠L) is as follows

There exists an ε>0 such that for every δ>0 and if 0<|x-a|<δ then |f(x)-L|>ε

Show that lim x->0 sin(pi/x)≠L

note: Take L=0.19

Q2) Calculate lim x->1 (sqrt(x+8)) and prove it by using ε and δ definition of limit.
• Oct 20th 2011, 11:20 AM
SammyS
Re: Plz help me about epsilon delta definition of limit.
Welcome to MHF.

Show us what you have tried.

Where are you stuck?
• Oct 20th 2011, 11:28 AM
mervecetin
Re: Plz help me about epsilon delta definition of limit.
Thank you!
about Q2 I find 0<|x-1|<δ and |sqrt(x+8)-3|<ε and |(sqrt(x+8)-3)*(sqrt(x+8)+3)/(sqrt(x+8)+3)|<ε equals |(x-1)/(sqrt(x+8)+3)|<ε Am I right

But about Q1 I can not find anything.
• Oct 20th 2011, 01:58 PM
Siron
Re: Plz help me about epsilon delta definition of limit.
Q2: That's a good beginning, to write it more formal, we have to show that:
$\forall \epsilon>0, \exist \delta>0, \forall x \in \ \mbox{dom f}: 0<|x-1|<\delta \Rightarrow |\sqrt{x+8}-3|<\epsilon$
Proof:
$|\sqrt{x+8}-3|<\epsilon \Rightarrow \left|\frac{(\sqrt{x+8}-3)(\sqrt{x+8}+3)}{\sqrt{x+8}+3}\right|\Rightarrow \left|\frac{x-1}{\sqrt{x+8}+3}\right| \Rightarrow \frac{|x-1|}{|\sqrt{x+8}+3|}<\epsilon$
Now, suppose $\delta=1$ therefore $|x-1|<1 \Leftrightarrow -1 ... go further with this given to find for each $\epsilon$ a corresponding $\delta$
• Oct 20th 2011, 02:02 PM
mervecetin
Re: Epsilon delta definition of limit.
• Oct 20th 2011, 07:20 PM
SammyS
Re: Epsilon delta definition of limit.
Quote:

Originally Posted by mervecetin
Q1) The precise definition of lim x->a (f(x)≠L) is as follows

There exists an ε>0 such that for every δ>0 and if 0<|x-a|<δ then |f(x)-L|>ε

Show that lim x->0 sin(pi/x)≠L

note: Take L=0.19

For Q1:

Start by inserting the given quantities, in this case L, a, and f(x), into the precise definition for the limit not existing.