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. **

lim*x*→0 (*x^*2 + 1) = 1 **In exercises 9–20, symbolically find ****δ ****in terms of ***ε***.** **15.**

lim*x*→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

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

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 $\displaystyle \delta$.