Berman functions

1) Which of these are infinitely large??
a) y=2^x sin^-1(sinx) as x-->+∞
b) y=(2+sinx)logx as x-->+∞
c) y=(1+sinx) logx as x-->+∞

2) Prove that if the limit of the function f(x) is equal to 'a' as x-->∞, then f(x) is representable as sum f(x) = a+g(x), where g(x) is is an infinitesimal as x-->∞.

3)The function u_n takes on the following values
u₁=1/4; u₂=1/4 + 1/10;...... u_n=1/(3+1) + 1/(3²+1) +.......+ 1/(3ⁿ+1)....

Prove that u_n tends to a certain limit as n-->∞

4) Prove the theorem - if the difference between two functions for one and the same variation of the independent variable is an infinitesimal, one of them increasing, the other decreasing at the same time, then both the functions tend to one and the same limit.

5) How many points of discontinuity (and of what kind) has the function y=1/log!x! ?? (!x!=mod x)

euclid , for 2nd i m not sure if it is correct

consider f(x) - g(x) .

Let this be equal to some k

so f(x) = g(x) + k

so g(x) = f(x) - k

now lim g(x) = 0
x->∞

so lim [ f(x) - k ] = 0
x->∞

so lim f(x) = k
x-> ∞

hence a = k

but qwerty hw did u get lim(x-->∞) g(x)=0 ??

that means the question is true for any tending value of x???

oops yes.. i din see that it was 1+sinx

as sinx can be anything between -1 and 1, for x-->∞, sinx can be -1 also so, 1+sinx = 0 (exact) (for some values of x)

so its only (b) as 2+sinx can never be equal to zero

similarly it wont be (a) cuz sin^-1(sinx) can be zero also

euclid is it given na that as x tends to infinity g(x) tends to zero