MATHEMATICAL ANALYSIS - we are fans
from the subject
(C) FRACTAL 2.03.2004
================================
Let's agree that we will talk about
mat. analysis. We will solve the type
howl calculation according to Kuznetsov number
first. But first let's agree
about notation.
x is the independent variable for
the functions under consideration.
f,g - functions of x.
n is a natural number.
oo - infinity.
lim - limit.
abs is the argument module.
c - affiliation.
A is a universal quantifier.
E is the existential quantifier.
Standard mathematical operations
ations like =,+,-,*,/,^, and also
elementary functions to denote
We'll be standard. Subscripts
We will write it in parentheses.
Further. A little theory. In the calculation
they require to know the following:
1. The concept of numerical sequence
lustfulness and its limits. Theorem about
boundedness of the convergent after-
validity.
Definition
a[n] is called a numeric sequence
attractiveness if
A (ncN) <- a[n]cR
Definition
The number a is called the limit of the number
basic sequence a[n],
if
(A e>0)(E n(e)cN)(A n>n(e)):
abs(a[n]-a) a[n]-limit.
2. The concept of the limit of a function is
chke. The concept of function is limited
noi in the vicinity of a point. Theoretically
MA about the limited function,
having a limit.
The definition is formulated in language
yke "epsilon-delta", and
Heine's definition makes it possible
nessdo not formulate indicated-
new theorem, since it is already
formulated for the limit of pos-
research.
3. Theorem on passage to the limit
in inequalities.
4. Theorem about the limit of intermediate
no function.
Well, this is all elementary.
5. The concept of continuity of a function
at the point. Prove continuity
functions cos(x).
Definition 1.
The function f is called continuous
at point x0, if
lim(f)=f(x0)
x->x0
One can also formulate
two definitions of continuity
(in epsilon-delta language and with
indicating all conditions: defining
validity, existence of limits
left and right).
Basic elementary, and traces
Consequently, the elementary functions
continuous in the region of their definition
leniya.
Let's prove that
(Ae>0)(Ed(e)>0)
(Ax:Abs(x-xO) inequality g you-
full. That. cos(x) function is not
is discontinuous by definition.
6. The first wonderful limit
lim(sin(x)/x)=1
x->0
7. Infinitesimal function and its
properties.
8. Theorems on arithmetic operations
erations over the limits (+,*,/).
And hence the continuity of the sum,
works, private. Continuous
the accuracy of a complex function.
9. Equivalent b.m. functions.
They are worth listing:
x->0
x~sin(x)~tg(x)~arcsin(x)~
~arctg(x)~ln(1+x)~e^x-11-cos(x)~x^2/2
a^x-1~x*ln(a)
Well, now you can get down to business
dachas
TASK 1.
---------
Prove that lim(a[n])=a
n->oo
(specify n(e)):
1.1. - a[n]=(3n-2)/(2n-1),a=3/2
Solution:
(Ae>0)(En(e)cN)(An>n(e)):
Abs((3n-2)/(2n-1)-3/2)
n(e)=(2e-1)/(4e-6)
Answer: n(e)=(2e-1)/(4e-6)
TASK 2.
---------
Calculate the numerical limit after-
duration:
2.7 -
lim[n->oo]
(((1+2n)^3-8n^3)/((1+2n)^2+4n^2)
Answer: 3/2 (this limit is
ratio of coefficients at sta-
higher powers of n, formally this is
can be shown by dividing by n^2).
TASK 3.
---------
Calculate the numerical limit after-
duration:
3.6 -
lim[n->oo]
(n^(6/5)-(27n^6+n^2)^(1/3))/
(n+n^(1/4))*(9+n^2)
Answer: -27
TASK 4.
---------
Calculate the numerical limit after-
duration:
4.2 -
lim[n->oo]
n(Sqrt(n(n-2))-Sqrt(n^2-3))
Answer: oo (multiply by conjugate)
the feminine is the sum of the roots, and then
according to the previous principle)
Share your thoughts about the article