Proving continuity

Jul 2015
603
15
United States
So, essentially, I am stuck on this problem. Note: I must solve problem using the definition I listed in question.

I will provide Picture of my work down below and the question.
 

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Jul 2015
603
15
United States
Please someone :( final is tomorrow morning. Just need help on this problem.
 

Plato

MHF Helper
Aug 2006
22,490
8,653
So, essentially, I am stuck on this problem. Note: I must solve problem using the definition I listed in question. I will provide Picture of my work down below and the question.
You are correct sofar. I will not do this for you but will gladly check your work.
You have absolute control over \(\displaystyle \delta\) but \(\displaystyle \varepsilon \) is a fixed positive number.
Look at what you have done:
\(\displaystyle \begin{align*}|f(x)-f(1)|&=\left|\frac{x-1}{x(\sqrt{x}+1)}\right| \\&=\frac{|x-1|}{|x(\sqrt{x}+1)|}\\&\le\delta\frac{1}{|x(\sqrt{x}+1)|} \end{align*}\)
Can you adjust \(\displaystyle \delta\) so as to make \(\displaystyle \frac{\delta}{|x(\sqrt{x}+1)|}<\varepsilon~~?\)
 
Jul 2015
603
15
United States
This is indeed the part I am stuck at...what to make delta.
 
Jul 2015
603
15
United States
so I am essentially trying to find out what to make delta, so epsilon can be greater than the difference of the absolute value of f(x) and f(1).
 
Jul 2015
603
15
United States
@Plato

I see a solution from my classmate, she put delta to equal min {1, sqrt(1/2)(1+sqrt(1/2))}

But this does not help me
 
Nov 2019
23
4
Boston
@Plato

I see a solution from my classmate, she put delta to equal min {1, sqrt(1/2)(1+sqrt(1/2))}

But this does not help me
If epsilon is small enough don't you think any specific number will not work for delta? It has to depend on epsilon! You need to express delta through epsilon, so that the inequality holds for any x.
Look at the denominator. What's the worst case scenario? I.e. when is the fraction at max?
 
Jul 2015
603
15
United States
okay, what if we set delta to equal epsilon/k where k is arbitrary. That will work, won't it not?
 
Nov 2019
23
4
Boston
okay, what if we set delta to equal epsilon/k where k is arbitrary. That will work, won't it not?
You need to prove that delta exists. I doubt involving arbitrary k accomplishes that. Try setting k to a particular value.