Could you help me with the problem?
Find delta using the definition of limits, given epsilon = 0,25
lim 1 / (2-x) = -1/3
x->5
Answer should be delta = 1
How can I get it?
Thanks.
No, the answer is not 1- but 1 is a perfectly valid answer.
The definition of "limit" says that, given any $\displaystyle \epsilon> 0$, there exist $\displaystyle \delta> 0$ such that if $\displaystyle |x- 5|< \delta$ then $\displaystyle \left|\frac{1}{2- x}- (-\frac{1}{3}\right|< \epsilon$.
Here, you are given that $\displaystyle \epsilon= 0,25$ so we want to have $\displaystyle \left|\frac{1}{2-x}+ \frac{1}{3}\right|< 0.25$
Go ahead and add the fractions on the left:
$\displaystyle \frac{3}{3(2-x)}+ \frac{2- x}{3(2-x)}=\frac{5- x}{3(2-x)}$
So we want $\displaystyle \left|\frac{5-x}{3(2-x)}\right|< .25$.
That is the same as $\displaystyle \left|5-x|\right|< \frac{3}{4}|x-2|$
Let's say that |5- x|< 1. Then -1< x- 5< 1 so 4< x< 6. then 2< x- 2< 4 so 2< |x-2|< 4. Certainly, $\displaystyle \frac{3}{4}(2)= \frac{3}{2}< \frac{3}{4}|x-2|$ then, so if we can make $\displaystyle |x- 5|< \frac{3}{2}$ we are done. Since we are already assuming $\displaystyle |x- 5|< 1$, and $\displaystyle 1< \frac{3}{2}$, it is sufficient to make $\displaystyle |x- 5|< 1$. That is, we can take $\displaystyle \delta= 1$.
Notice that this is not saying that $\displaystyle \delta$ has to be equal to 1. Obviously any smaller number would work and there might be some larger value that would work. But $\displaystyle \delta= 1$ is sufficient.
Had I said "Let's say that |5- x|<" some other number, I would have gotten an answer other than 1, but equally valid.