The document discusses limits and continuity of functions. It defines the limit of a function as x approaches a number, and gives examples of limits that do and do not exist. It then discusses properties of limits, including: - If limits of f(x) and g(x) both exist as x approaches a, then the limit of their sum and product also exist. - If the limit of f(x) exists as x approaches a, then the limit of kf(x) is k times the limit of f(x) for any constant k. Finally, it discusses continuity, defining a function to be continuous at a point if the limit exists there and equals the function value. It gives

2. The number Lis called the limit of the function
f(x) as x approaches a, written

3. x
x

4. x 0 x

6. lim h x 2
x 1
lim h x 1
x 1
lim h x does not exist
x 1

8. ) If f(x) = c ( a constant function), then
for any a
x a
n n
) where n is a positive
x a
integer

9. ) If f x and gx
x a x a
exist , then
x a x a x a

10. ) If lim f ( x ) and g x
x a x a
exist , then
x a x a x a

11. - If lim f ( x ) exists, then for any
x a
constant k,
lim k f ( x )] k lim f ( x )
x a x a

12. lim f ( x ) gx
x a x a
f (x ) lim f ( x )
x a
lim
x a g (x ) lim g ( x )
x a

13. ) If lim f ( x ) exits, and n is a positive
x a
integer
n
n
x a x a

14. If
does not exist, since

16. ) 1
lim p 0 where p > 0
x x
1
lim p 0 where p > 0
x x

19. The limit does not exist

20. If f(x) is a rational function and is the
term with greatest power in the numerator
and is the term with greatest power in
the denominator, then

23. A function f(x) is continuous at a point b if and
only if:
) f(x) is defined at x = b
) lim f(x) exist
x b
) ) lim f(x) = f(b)
x b

24. Show that f(x) = / (x- ) is continuous at x=
and discontinuous at x =
f( )= / lim x f(x) = / =f( )
At x= the function is not defined.

25. The function is not defined at x=-
So it is not continuous at x=- , but
the limit exist.

27. A polynomial function is continuous at all
points of its domain.

28. Find all points of discontinuity of
f(x) = x - x +
X + X-
X + X - = (x+ ) (x- ).
The denominator is zero when x=- or x=
x=-
Thus the function is discontinuous at x=-
and x= only.