7.2. EXERCISES 131

11. Find limh→0(x+h)3−x3

h

12. Find limh→01

x+h−1x

h

13. Find limx→−3x3+27x+3

14. Find limh→0

√(3+h)2−3

h if it exists.

15. Find the values of x for which limh→0

√(x+h)2−x

h exists and find the limit.

16. Find limh→03√

(x+h)− 3√xh if it exists. Here x ̸= 0.

17. Suppose limy→x+ f (y) = L1 ̸= L2 = limy→x− f (y) . Show limy→x f (x) does not ex-ist. Hint: Roughly, the argument goes as follows: For |y1− x| small and y1 > x,| f (y1)−L1| is small. Also, for |y2− x| small and y2 < x, | f (y2)−L2| is small. How-ever, if a limit existed, then f (y2) and f (y1) would both need to be close to somenumber and so both L1 and L2 would need to be close to some number. However,this is impossible because they are different.

18. Suppose f is an increasing function defined on [a,b] . Show f must be continuousat all but a countable set of points. Hint: Explain why every discontinuity of f is ajump discontinuity and

f (x−)≡ limy→x−

f (y)≤ f (x)≤ f (x+)≡ limy→x+

f (y)

with f (x+) > f (x−) . Now each of these intervals ( f (x−) , f (x+)) at a point, xwhere a discontinuity happens has positive length and they are disjoint. Furthermore,they have to all fit in [ f (a) , f (b)] . How many of them can there be which have lengthat least 1/n?

19. Let f (x,y) = x2−y2

x2+y2 . Find limx→0 (limy→0 f (x,y)), limy→0 (limx→0 f (x,y)) . If youdid it right you got −1 for one answer and 1 for the other. What does this tell youabout interchanging limits?

20. The whole presentation of limits above is too specialized. Let D be the domain of afunction f . A point x not necessarily in D, is said to be a limit point of D if B(x,r)contains a point of D not equal to x for every r > 0. Now define the concept of limitin the same way as above and show that the limit is well defined if it exists. That is,if x is a limit point of D and limy→x f (x) = L1 and limy→x f (x) = L2, then L1 = L2. Isit possible to take a limit of a function at a point not a limit point of D? What wouldhappen to the above property of the limit being well defined? Is it reasonable todefine continuity at isolated points, those points which are not limit points, in termsof a limit as is often done in calculus books?

21. If f is an increasing function which is bounded above by a constant M, show thatlimx→∞ f (x) exists. Give a similar theorem for decreasing functions.