1. ## endless solutions question

2. Originally Posted by transgalactic
Im sorry...could you please try to reword this in a more comprehensible way? You may be able to understand what you mean...but to me its just a bunch of words. I cant tell whats part of the question and whats part of your question...you say words like can but I think you mean can't. So we of course will be glad to help...and myabe someone will better understand you than me, but you may want to reword this just in case. Also, you may want to tell us exactly what you don't understand and why so we can better help you.

P.S. Im not trying to nag, Im just trying to make sure you get the help you deserve and expect from this site

3. i need to prove that there is an endless amount of x's satisfying
x=tanx
for that they take some index k
and on the interval of (x_k,x_k+1)
and they construct on this specific interval
two limits.

on the one they get a positive value
an on the other a negative value
so between them they must have on solution
because k could be anywhere
there is endless solutions

my problem is that i cant how is it possible to have such interval
(x_k,x_k+1)
because there are no members in it

and i cant understand how they solved those limit
i cant see how they got +infinity on one
and - infinity on the other
??

4. To understand this proof, you need to think about the graph of the tan function (see attachment). For each integer k, as x goes from $\displaystyle \bigl(k-\tfrac12\bigr)\pi$ to $\displaystyle \bigl(k+\tfrac12\bigr)\pi$, tan(x) goes from –∞ to +∞. At some point in that interval, the graph of tan(x) must cross from being below the line y=x to being above that line. Therefore the equation tan(x)=x has a solution in that interval.

If you keep that picture in mind as you read the proof, you should be able to make sense of it.