There is an integer, which is added, subtracted, multiplied and divided. The sum, difference, product and quotient are added up to 81. What is this integer?

Suppose the integer is X
Add up to 2x
Subtract to 0
Multiply to get x * x
Divide by 1
So we get the univariate quadratic equation: 2x + 0 + X * x + 1 = 81
That is, the square of (x + 1) is 81
So x = 8

If an > BN and the limit of an = a, and the limit of BN = B, then a > b, for example

If an > BN and the limit of an = A and the limit of BN = B, then a > b, right? Take a counter example
Only a ≥ B can be obtained, which is not necessarily strictly greater than
For example: a [n] = 1, B [n] = 1-1 / 2 ^ n
a[n]>b[n]
lima[n]=limb[n]=1.

Given that the limit of [5N - √ (an ^ 2-BN + C)] is 2, find the values of a and B It is known that the limit of [5N - √ (an ^ 2-BN + C)] is 2, that is, Lim [5N - √ (an ^ 2-BN + C)] = 2. Find the values of a and B. why? (√ is the root sign) That is, LIM (n →∞) [5N - √ (an ^ 2-BN + C)] = 2

From the original formula, we get
lim(5n)-lim√(an ²- bn+c)=2
lim(5n-2)=lim√(an ²- bn+c)
According to the uniqueness of the limit, we get
5n-2=√(an ²- bn+c)
Namely: (5n-2) ²= an ²- bn+c
25n ²- 20n+4=an ²- bn+c
If the polynomials are equal, the coefficients are equal,
So a = 25, B = 20

Given {an}, {BN} satisfies LIM (2An + BN) = 1, LIM (an-2bn) = 1, find the value of LIM (anbn)

Let LIM (an) = x, LIM (BN) = y
2x+y=1
x-2y=1
therefore
x=3/5,y=-1/5
lim(anbn)=xy=-3/25

If LIM (an / BN) = a (a is not 0) LIM (an) = 0, it is proved that LIM (BN) = 0 It can be proved by the definition of sequence limit

To prove the contrary, it is assumed that LIM (BN) is not equal to 0, LIM (an / BN) = LIM (an) / LIM (BN) = 0, which is inconsistent with the meaning of the question, so LIM (BN) = 0

If LIM (square of an + bn-5) / (2n + 1) = 1, why a = 0, otherwise the limit does not exist

There are two possibilities for this question, one is that the denominator can be reduced, the other is that it cannot be reduced. The first case: if it is reduced, the limit exists: if it cannot be reduced, it is assumed that the numerator is of second order and the denominator is of first order. The limit exists only when the order of the numerator is less than or equal to the order of the denominator

There is an integer, which is added, subtracted, multiplied and divided. The sum, difference, product and quotient are added up to 81. What is this integer?