uchihajeff said:
-1<x+2<1, -2<x+1<0
|x+1|<0
|x+2|< e * |x+1|
This part is wrong, you are taking the absolute value of x + 1, and you have -2 < x + 1 < 0, you should have 0 < |x + 1| < 2, not |x + 1| < 0
pardesi said:
...
|f(x)-b| < \epsilon for all real x satisfying |x-a| < \delta
Well, nope, x cannot be a, you should add '0 <' part in front of it, like this:
0 < |x - a| < \delta \Rightarrow |f(x) - b| < \epsilon. :)
Yup, as
pardesi has pointed out, you should re-arrange the expression, so that you could obtain:
|f(x) - b| < ... < k |x - a| < epsilon (where k is any constant, this is the tricky part, you should find the appropriate k)
~~~> |x - a| < epsilon / k
So we can choose delta to be (epsilon / k), so, for:
x : 0 < |x - a| < \delta = \frac{\epsilon}{k} \Rightarrow |f(x) - b| < ... < k |x - a| < k \epsilon < k \times \frac{\epsilon}{k} = \epsilon as required.
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I'll help you do the first problem, and give you some hints for the seconds one, they are pretty much the same. :)
Problem 1:
Prove:
\lim_{x \rightarrow -2} \frac{1}{x + 1} = -1
So, we'll start here:
\left| \frac{1}{x + 1} + 1 \right| = \left| \frac{x + 2}{x + 1} \right| = \textcolor{blue}{\left| \frac{1}{x + 1} \right|} \textcolor{red}{|x + 2|}.
Now, we will obtain the
k from the blue part. We will find out the
maximum value for |1 / (x+1)|, and assign k to be that value.
To find the maximum value for |1 / (x+1)|, we'll make the restriction:
\textcolor{green} {0 < |x + 2| < \delta \leq 0.5} \Rightarrow 0 < |x + 2| < 0.5 \Rightarrow -0.5 < x + 2 < 0.5
\Rightarrow -1.5 < x + 1 < -0.5 \Rightarrow 0.5 < |x + 1| < 1.5 \Rightarrow \frac{2}{3} < \frac{1}{|x + 1|} < 2.
Ok, so we have:
\left| \frac{1}{x + 1} + 1 \right| = \textcolor{blue}{\left| \frac{1}{x + 1} \right|} \textcolor{red}{|x + 2|} < \textcolor{blue}{2} \textcolor{red}{|x + 2|}, ok, so we choose:
\delta = \mbox{min} \left( 0.5 , \frac{\epsilon}{2} \right).
I think your textbook's answer is, somehow,
mistyped.
Ok, let's think about it, in the
green part above (I mean, the restriction part), why didn't I choose: 0 < |x + 2| < \delta \leq 1? I didn't choose 1, that's
not appropriate in this case. Can you see why?
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So, now, hint for second problem. :)
We must rearrange the expression to give k |x - a|, i.e, we must have |x - a| in our expression, so, multiply by its conjugate, we obtain.
|\sqrt{x} - 2| = \frac{\textcolor{red}{|x - 2|}}{\textcolor{blue}{| \sqrt{x} + 2 |}}
Can you go from here? :)
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EDIT: I have corrected the last LaTeX code for like 3 times, but it still shows the wrong image. >"<
Ok, now, LaTeX seems to work normally again. ^.^