# Determining a δ

• Sep 4th 2009, 08:53 AM
Rker
Determining a δ
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.
limx0 (x^2 + 1) = 1

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

15.
limx1 (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?
• Sep 4th 2009, 10:11 AM
VonNemo19
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.
limx0 (x^2 + 1) = 1

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

15.
limx1 (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

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

To do this we must have

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

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

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

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

$\displaystyle x^2<0.1$

Can you see how to find delta?

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