2.5. A quick method for finding limits of
rational
functions as
or
Let f (x) be a polynomial and let axn be its term of highest degree. Let g(x) be another polynomial and let bxm be its term of highest degree.
Then
and
Example: Evaluate the following limits:
a)
;
b)
;
c)
Solution: By the preceding observations,
a)
=
=
;
b)
=
=
;
c)
=
.
Exercises
In exercises 1-5 use definition of limit to prove that the given limit statement is correct.
1.
2.
3.
4.
5.
In exercises 6-20 find the limits.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
Answers
6. 3; 7. 157; 8. 2; 9. 0; 10. 4/5; 11. 5/11; 12. Does not exist; 13. –4/5; 14. 4/3; 15. 5/3; 16. ; 17. ; 18. - ; 19. 0; 20. - .
2.6. Limits involving radicals
Example:
Find
Solution:
Using the property of limit
=
=
=
.
Example: Find:
a)
;
and b)
.
Solution:
Before
beginning the solution, note that if x
is positive,
,
but if x
is negative,
.
a)
=
=
=
=
=
.
b)
=
=
=
=
.
Example:
Find
Solution:
Let us substitute z6=1+x. We choose z6 in order to take roots easily. From substitution it is easy to see that if x0 then z1. Using substitution, we get
=
=
= =
=
=2/3.
Example:
Find
.
Solution:
When
we get
.
Let us multiply and divide given function to
.
We will get
=
=
=
=
=
=
=
=
=3.
Example:
Find
.
Solution:
As
,
the values of x are
eventually positive, so we can replace
by x
where desirable. We obtain
=
=
=
=
=1/3.
Exercises
In exercises 1-21 find the limits.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
Answers
1.
1/4; 2.
5/6; 3.
0; 4.
–1/2; 5.
0; 6.
;
7.
5/2; 8. –3/2;
9.
a/2;
10.
–40; 11.
1; 12.
–1/2; 13.
0; 14.
-
;
15.
;
16.
;
17.
does not exist; 18.
6; 19.
4; 20.
6; 21.
3/4.
2.7. One sided limits
Let
.
If x approaches
0 from the right, f
(x)
is always 1. If x
approaches 0 from the left, f
(x)
is always –1. This introduces the notion of one-sided limits.
Definition: Right-hand limit of f (x) at a. Let f be a function and
a some fixed number. Assume that the domain of f contains an open interval (a, b). If, as x approaches a from the right, f (x) approaches a specific number L, then L is called the right-hand limit f (x) as x approaches a.
This is written:
or as x
a+,
f (x)L.
The assertion that is read : “ the limit of f as x approaches a from the right is L”, or “ as x approaches a from the right, f (x) approaches L”.
The
left-hand limit is defined similarly. The only differences are that
the domain of f
must contain an open interval of the form (c,
a)
and f (x)
is examined as x
approaches a
from the left. The notations for the left-hand limits are: 
or as x
a-,
f (x)L.
;
;
;
.
Example:
Find: a)
; b)
Solution: In both examples the limit of the numerator is –2 and denominator is 0, so the limit of the ratio does not exist. We need to analyze the sign of the ratio. As x approaches 4 from the right, the ratio is always negative, and as x approaches 4 from the left the sign of the ratio is eventually positive (after x exceeds 2), so
and
Example:
Find
Solution:
As x
approaches -1 from the right, (x+1)
approaches 0 from right. The reciprocal
stays positive and increases beyond all bounds.
Example:
Find: a)
and b)
Solution:
a) When x
approaches 0 from the left, then
is
large and negative. We can write: 
.
Since
is large when
,
then
is also large, and its reciprocal
is
a small number. Consequently
b) When x
approaches 0 from the right, then
is
large and positive. The value
will be large and positive.
.