Determining a δ

Quote:

Originally Posted by Rker

I have absolutely no idea how to solve these type of problems. My teacher gave a lecture about this subject two days ago, and I took a look at this stickied thread, but I'm still stuck. :s

In exercises 1–8, numerically and graphically determine a δ corresponding to (a) ε = 0.1 and (b) ε = 0.05. Graph the function in the ε − δ window [x-range is (a − δ, a δ) and y-range is (L − ε, L + ε)] to verify that your choice works.

1.
limx→0 (x^2 + 1) = 1

In exercises 9–20, symbolically find δ in terms of ε.

15.

limx→1 (x^2 + x − 2)/(x − 1) = 3

52.

A fiberglass company ships its glass as spherical marbles. If the volume of each marble must be within ε of π/6, how close does the radius need to be to 1/2?

For 1.

You wish to show that

$\lim_{x\to0}(x^2+1)=1$ .

To do this we must have

$|f(x)-L|$ whenever $|x-a|<\delta$ .

So, given that $\epsilon=0.1$ we proceed

$|(x^2+1)-1|<0.1$

$|x^2|<0.1$ . Since $x^2>0$ for all x,

$x^2<0.1$

Can you see how to find delta?

PS Finding a delta graphically is easy. Just draw the graph. then draw the lines $L+\epsilon$ and $L-\epsilon$ . Where those lines intesect the graph, draw vertical lines down to the x-axis. the line which is closest to $x=a$ is $\delta$ .