Epsilon Delta Proof, need clarification

[haribol]

Prove that

<br /> lim_{x \rightarrow c} \ \ \frac{1}{x}=\frac{1}{c} \ ,c\neq0<br />

Proof

We must find \delta such that:

1.

<br /> 0&lt;|x-c|&lt;\delta \ \Rightarrow \ | \ \frac{1}{x}=\frac{1}{c}|&lt;\epsilon<br />

Now,

2.

| \ \frac{1}{x}=\frac{1}{c}|=| \ \frac{c-x}{xc}|= \frac{1}{|x|} \cdot \frac{1}{|c|} \cdot |x-c|<br />

The factor \frac{1}{|x|} is not good if its near 0. We can bound this factor if x can be away from 0. Note:

3.

<br /> |c|=|c-x+x| \leq |c-x|+|x|<br />

so

4.

<br /> |x| \geq |c|-|x-c|<br />

Thus if we choose

5.

<br /> \delta \leq \frac{|c|}{2}<br />

6.

then we can succeed in making
<br /> |x| \geq \frac{|c|}{2}<br />

Finally, if we also require

7.

\delta \leq \frac{\epsilon c^2}{2}

then,

8.

<br /> \frac {1}{|x|} \cdot \frac{1}{|c|} \cdot |x-c| &lt; \frac {1}{|c|} \cdot \frac {1}{|c|/2} \cdot \frac{\epsilon c^2}{2} = \epsilon<br />


My questions

1. On step 2, its said to "bound" the factor 1/|x|. What does this mean?

2.How did the |c-x| on step 3 went to |x-c| on step 4?

3. How did they choose \delta to be that value?

Any help on this subject is very much appreciated, thank you in advance.

This example is from "Calculus 8th Edition" by Varberg, Purcell and Rigdon.

 

[arildno]

1. Since c doesn't equal 0, there exists a (tiny) region about c which doesn't contain 0. This region is what we care about.
2. The negative of the first difference has equal magnitude as the difference itself.

 

[phoenixthoth]

Some more thoughts...

1. As Arnildo wrote, as c!=0, there is an interval
I:=[c-z,c+z]
that doesn't contain the origin. If you like, z can be half the distance between c and 0 to ensure
a. if c<0 that c+z<0
b. if c>0 that c-z>0.

z could also be 3/4 the distance between 0 and c or some such. A picture might be helpful.

On I, 1/|x| can be "bounded" which means that we can find a number M such that for x in I, 1/|x| <= M. You can find a formula for M depending on c and z but NOT x. This M is used in step 8 somewhere.

You have |1/x - 1/c| < (M/|c|) |x-c| < (M/|c|) delta.

Then if delta = epsilon * (|c|/M), then we'd have that for an epsilon>0 there is a delta such that if |x-c|<delta, then |1/x-1/c|<epsilon.

 

[haribol]

Thank you very much guys