# Proving continuity

#### math951

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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Here is my work

#### math951

Please someone final is tomorrow morning. Just need help on this problem.

#### Plato

MHF Helper
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 $$\displaystyle \frac{\delta}{|x(\sqrt{x}+1)|}<\varepsilon~~?$$

#### math951

This is indeed the part I am stuck at...what to make delta.

#### math951

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).

#### math951

@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

#### Lev888

@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?

#### math951

okay, what if we set delta to equal epsilon/k where k is arbitrary. That will work, won't it not?

#### Lev888

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.