2. Consider the function with the rule = \frac{x^n}{1+x^n})
where 
is a positive even integer.
a) show that  = 1 - \frac{1}{x^n+1})
, this i have done, just long divide and there's the answer. The next part i don't quite understand.
b) Show that  < 1)
for all x
Much thanks for any help!
If
is a positive even integer, then
is always positive for any real value of
. Additionally, it makes the function even:
 = 1 - \frac{1}{(-x)^n + 1} = 1 - \frac{1}{(-1)^nx^n + 1} = 1 - \frac{1}{x^n + 1} = f(x))
Hence,
)
is symmetrical about the y-axis, which means that we only need to observe the behaviour of the function for one side of the y-axis, since everything is a mirror image on the other side. I'll choose the positive side:
.
At
,
 = 1 - \frac{1}{0+1} = 0)
.
Then, as
,
 \rightarrow 1 - 0 = 1)
, since
decays to zero as
approaches infinity. Hence, the function asymptotically approaches 1.
We can prove the monotonicity (always moving in one direction, no turning points) of this function for
, which is necessary to show that nothing funny happens in between zero and infinity. However, it should be obvious enough that the decay of the fraction only goes one way, and hence I'll call it a trivial exercise. To do it analytically, find
)
and show that it is either always positive or equal to zero (increasing monotonicity), or always negative or equal to zero (decreasing monotonicity) for
.
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