- •2.1. Definition of limit
- •2.2. Computations of limits
- •2.3. Limits of polynomials as or
- •2.4. Limits of rational functions as or
- •2.5. A quick method for finding limits of
- •2.6. Limits involving radicals
- •2.7. One sided limits
- •2.8. Existence of limits
- •2.9. Continuity
- •2.10. The limit of trigonometric functions.
- •2.11. The number e. Second remarkable limit
2.2. Computations of limits
Let f and g be two functions and assume that
and both exist. Then
1.
2.
3. , for any constant k
4.
5. , if
6. if
7. if
Example: Find
Solution: = - + =
= -4 + =52-45+3=8
Example: Find
Solution: The numerator and denominator both have a limit of zero as x approaches 3, so there is a common factor of (x-3). We proceed as follows:
= = =0.
Example: Find
Solution: The numerator and denominator both have a limit of zero as x approaches -4, so there is a common factor of (x-(-4))=x+4. We proceed as follows:
= = =-2/7.
Example: Find
Solution:
= =
= = =
= =6/3=2.
Example:
Find
Solution: The numerator and denominator both have a limit of zero as x approaches 4. Let us rewrite x2-6x+8 using x2+px+q=(x-x1)(x-x2) where x1 and x2 are roots of equation x2+px+q=0. An equation
x2-6x+8=0 has roots x1=2 and x2=4, so x2-6x+8=(x-2)(x-4).
After substituting we get
= = =4-2=2.
Example:
Find
Solution: Again when x=1 both numerator and denominator have a limit of zero. Let us factor given function using identity
ax2+bx+c=a(x-x1)(x-x2)
where x1 and x2 are roots of equation ax2+bx+c=0. The limit can be obtained as follows:
= = =
= =5/3.
2.3. Limits of polynomials as or
Sometimes it is useful to know how f (x) behaves when x is a large positive (or a negative number of large absolute value).
Rather than writing “as x gets arbitrary large through positive values, f (x) approaches the number L”, it is customary to use the notation . This is read “as x approaches infinity, f (x) approaches L”, or “the limit of f (x) as x approaches infinity is L”.
Similarly, means that “the limit of f (x) as x approaches minus infinity is L”.
Remark: Notations and are equivalent to each other, and we will use both of these notations.
Remark: All properties of limits stated above hold when x a is replaced by
or by .
, n=1,2,3,……..;
;
;
.
A polynomial P(x) = c0+c1x+c2x2+……..+cnxn behaves like its term of highest degree as or . If cn0, then
.
Example: Find
Solution: = .
Example: Find
Solution: = .
Remark: It is important to keep in mind that “ ” or “ ” is not a number. The last two limits above do not exist.
2.4. Limits of rational functions as or
To find limits of rational functions as or we divide the numerator and denominator of a rational function by the highest power of x that occurs in the denominator. What happens then depends on the degrees of the polynomials involved.
Example: Find
Solution: Divide the numerator and denominator by the highest power of x that occurs in the denominator; this is x1=x.
We obtain
= = =
= = .
Example: Find
Solution: Divide the numerator and denominator by the highest power of x that occurs in the denominator, namely x3. We obtain
= =
= .
Example: Find
Solution: Divide the numerator and denominator by x to obtain
= ,
since 7x2-4x + , , and as .