Prove 2 Limits Questions with Definition 2.4.1

  • Thread starter Thread starter uchihajeff
  • Start date Start date
  • Tags Tags
    Limit
uchihajeff
Messages
10
Reaction score
0

Homework Statement


Use Definition 2.4.1 to prove that the stated limit is correct.

Definition 2.4.1 in my book is:

lim as x->a of f(x) = L

if given any number e(epsilon)>0 we can find a number d(delta)>0 such that

|f(x)-L|<e if 0<|x-a|<d​

Homework Equations


Question 31. lim as x>-2 of 1/(x+1) = -1
Question 33. lim as x>4 of sqrt(x) = 2

The Attempt at a Solution


31. |1/(x+1) + 1|<e, 0<|x+2|<d
|(x+2)/(x+1)|<e
set d<=1
-1<x+2<1, -2<x+1<0
|x+1|<0
|x+2|< e * |x+1|
...then I get stuck

32. |sqrt(x)-2|<e, 0<x-4<d
sqrt(x)<e-2
x<(e-2)^2
x-4<(e-2)^2-4
...by here I'm probably already wrong
d=(e-2)^2-4

4. The answers in the back of the book
31) d=min(1,e/(1+e))
33) d=2e

P.S. Sorry, I don't know how to use Latex or whatever mathematical typing system you guys use here, so it's a little messy/unreadable.
Thanks in advance for the help!
 
Physics news on Phys.org
well one of the best way and probably only way to prove limits using the absic definition is to do like thsi

suppose i need to prove

\lim _{x/to/a} f(x)= b
that means i have to prove for every positive real number \epsilon there exists a positive delta such that

|f(x)-b| &lt; \epsilon for all real x satisfying |x-a| &lt; \delta

for a moment assume that there exists a x such that
|f(x)-b| &lt; \epsilon
manipulate such that u reach a stage where k|x-a| &lt; \epsilon
for some constant k then ur \delta is
k\epsilon
here the best way i would prefer is u looking into the proof of the fact that
\lim_{x \to a} \frac{f}{g}=\frac{\lim_{ x \to a} f}{\lim_{x \to a} g}...
and similarly for the second part
\lim_{x \to a} f^{\frac{1}{n}}={\lim f }^{\frac{1}{n}}...
 
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| &lt; \epsilon for all real x satisfying |x-a| &lt; \delta

Well, nope, x cannot be a, you should add '0 <' part in front of it, like this:
0 &lt; |x - a| &lt; \delta \Rightarrow |f(x) - b| &lt; \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 &lt; |x - a| &lt; \delta = \frac{\epsilon}{k} \Rightarrow |f(x) - b| &lt; ... &lt; k |x - a| &lt; k \epsilon &lt; k \times \frac{\epsilon}{k} = \epsilon as required.

---------------------------------

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 &lt; |x + 2| &lt; \delta \leq 0.5} \Rightarrow 0 &lt; |x + 2| &lt; 0.5 \Rightarrow -0.5 &lt; x + 2 &lt; 0.5

\Rightarrow -1.5 &lt; x + 1 &lt; -0.5 \Rightarrow 0.5 &lt; |x + 1| &lt; 1.5 \Rightarrow \frac{2}{3} &lt; \frac{1}{|x + 1|} &lt; 2.

Ok, so we have:
\left| \frac{1}{x + 1} + 1 \right| = \textcolor{blue}{\left| \frac{1}{x + 1} \right|} \textcolor{red}{|x + 2|} &lt; \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 &lt; |x + 2| &lt; \delta \leq 1? I didn't choose 1, that's not appropriate in this case. Can you see why?

--------------------------------------

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? :)

--------------------------------------

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. ^.^
 
Last edited:
VietDao29 said:
Well, nope, x cannot be a, you should add '0 <' part in front of it, like this:
well it is not necessary that x be b but there is no problem if it hiolds true for b infact then it becomes continious at that point:smile:
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
Back
Top