From this we see that the critical set for is . We can determine the sign of by looking at the signs of its factors: Since is odd, I will consider only points where .

Thus is strictly increasing on and on , and is strictly decreasing on . Also

We see that is unbounded on any interval or , since the numerator of the fraction is near to , and the denominator is near to on these intervals. Also

so is large when is large. ( is the product of and a number near to .) Using this information we can make a reasonable sketch of the graph of .

Here has a local maximum at and a local minimum at . It has no global extreme points.

if for every there is an such that for all .

We say

if for every there is an such that for all . Let be a real valued function such that , and let . We say

if contains an interval and for every sequence in

We say

if contains an interval and for every sequence in

Similar definitions can be made for

We say if contains some interval and for every sequence in

Similarly if we can define

and

Also,

and

The situation here is very similar to the situation in the case of ordinary limits, and we will proceed without writing out detailed justifications.

Hence is a critical point for if and only if . The critical points of in are thus and , and the critical points in are . Now and

and . Also note . Since is continuous on , we know that has a maximum and a minimum on this interval, and since for all , the maximum (or minimum) of on will be a global maximum (or minimum) for . Since is differentiable everywhere, the extreme points are critical points and from our calculations has a maximum at and a minimum at . I will now determine the sign of on :

Thus is strictly increasing on and is strictly decreasing on . We can now make a reasonable sketch for the graph of .

**a)**- .
**b)**- .
**c)**- .
**d)**- .

Thus,