prove that for c<0
there is only one solution to
$\displaystyle xe^{\frac{1}{x}}=c$
??
i tookk the left side as f(x)
for x=1 we have f(1)>0
the limit as x->-infinity is -infinity
what to do?
?
prove that for c<0
there is only one solution to
$\displaystyle xe^{\frac{1}{x}}=c$
??
i tookk the left side as f(x)
for x=1 we have f(1)>0
the limit as x->-infinity is -infinity
what to do?
?
At this point, do you even know what you are talking about?
$\displaystyle \delta/\varepsilon$ has absolutely nothing to do with it, as far as I can tell.
If $\displaystyle c<0$ then $\displaystyle 2c<c<\frac{c}{2}$ then because $\displaystyle f$ is increasing $\displaystyle f(2c)<f(c)<f\left(\frac{c}{2}\right)$.
ok i see it now .
if its decreasing we take smaller values and they get a bigger function value
so from the decreasing property we can directly write
$\displaystyle f(2c)<f(c)<f\left(\frac{c}{2}\right)$
from here its straight forward indermidiate value theorem
so why we need the limit
when x goes to zero the limit is sero
?
i need to show that there is x1 f(x1)<c
f(x2)>c
from the limit when x goes to sero we get zero soby limit definition -e<f(x)<e
from the limit when x goes minus infinity f(x)<-N
what e to chhose?
what N to choose?
As stated, the problem is to prove that the equation $\displaystyle xe^{1/x}=c$ has 0 or 1 solution for each c < 0. That's what "only one" means: there cannot be more than one. If the problem said "there exists one solution to this equation," that would mean "at least one, but possibly more." To stress that there exists exactly one solution, people sometimes say, "there exists one and only one solution."
Now, this interpretation is arguable and possibly not every everybody would agree that "only one" means <= 1 and "one" means ">= 1." I would therefore avoid statements containing "only one." However, since it is given, the default interpretation of this problem for me is to prove that there cannot be two solutions.
So, what exactly do you need to prove here? You may need to contact your instructor for clarification.