# Thread: Need help finding δ in this calculus problem!

1. ## Need help finding δ in this calculus problem!

Hi all,

The problem shows the graph of f(x) = x^2 - 1 and asks to find δ such that if 0 < |x - 2| < δ, then |f(x) - 3| < 0.2.

This is in the first chapter of my calculus book and dealing strictly with limits but I don't know how to do the problem, anyone want to chime in?

Thanks!

2. ## Re: Need help finding δ in this calculus problem!

Originally Posted by Odonsky
The problem shows the graph of f(x) = x^2 - 1 and asks to find δ such that
if 0 < |x - 2| < δ, then |f(x) - 3| < 0.2.
First take note that $|f(x)-3|=|x-2|\;|x+2|$. HOW & WHY?

Note that the factor $|x-2|$ is the one we control with the $\delta~.$

Work your way through the following:
$\\|x-2|<1~\Rightarrow\\-1

So the other factor $|x+2|$ is less that five if we start with $\delta<1$.

What would happen if $\delta=\frac{1}{25}~?$

Well $\delta<1$ so $|f(x)-3|=|x-2|\;|x+2|<\frac{1}{25}\cdot 5<~?$

3. ## Re: Need help finding δ in this calculus problem!

Originally Posted by Plato
First take note that $|f(x)-3|=|x-2|\;|x+2|$. HOW & WHY?

Note that the factor $|x-2|$ is the one we control with the $\delta~.$

Work your way through the following:
$\\|x-2|<1~\Rightarrow\\-1

So the other factor $|x+2|$ is less that five if we start with $\delta<1$.

What would happen if $\delta=\frac{1}{25}~?$

Well $\delta<1$ so $|f(x)-3|=|x-2|\;|x+2|<\frac{1}{25}\cdot 5<~?$
To answer the first question as to why |f(x) - 3| = |x - 2| |x + 2| I really do not know why, so the rest of it does not make sense either.

4. ## Re: Need help finding δ in this calculus problem!

Originally Posted by Odonsky
To answer the first question as to why |f(x) - 3| = |x - 2| |x + 2| I really do not know why, so the rest of it does not make sense either.
If that is true, then you are clearly not prepared for a course in calculus.

As a retired chair of a university mathematics department, I advise you to do a good course in Pre-Calculus.
You need a good review of the properties of absolute value and inequalities. I suspect you could benefit in many more ways from such a course.

5. ## Re: Need help finding δ in this calculus problem!

Surely you can at least evaluate f(x) - 3 and factorise it...

6. ## Re: Need help finding δ in this calculus problem!

Originally Posted by Plato
If that is true, then you are clearly not prepared for a course in calculus.

As a retired chair of a university mathematics department, I advise you to do a good course in Pre-Calculus.
You need a good review of the properties of absolute value and inequalities. I suspect you could benefit in many more ways from such a course.
I am in the process of preparing for an upcoming summer calculus course and have never taken this subject before, hence my inability to solve the above problem set.
I'm trying to self-study in the meantime using the internet.

7. ## Re: Need help finding δ in this calculus problem!

Originally Posted by Prove It
Surely you can at least evaluate f(x) - 3 and factorise it...
What I know about evaluating a function is to substitute x with whatever it equals but in this case I do not know what x equals.