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first we should know that 0.999...... is not a number.
Note
let sinx=t for all x in [0,pi/2],then we got cosx=sqrt(1-x^2).so,(sinx)^25/[(sinx)^25+(cosx)^25]=t^25/[t^25+(\sqrt (1-x^2))]25=f(t).we can get the extreme values of f(t) as follows:
1.find f'(t);
2.find t such that f'(t)=0;for example t1,t2;
3.f(t1),f(t2) is the maxium and inum
4.estimate the integral by the extreme value.
we call logarithm function with base 10 common logarithm and it was denoted by lgx. TheDude had gave you the right answer.
since sin(1/x) is bounded so there exist a number M such that |sin1/x|<M,then |x*sin(1/x)|<M|x|
so it approaches 0 as x approaches 0. in fact we can let M=2.
to calcute area between some functions. please follow these steps:
step1: give their graphs togher.
step2: describe the area,remember we have two type of describtions called x-type and y-type.
step3: transffer the area to integration.
step4: calcute the integration.
∞ is not a number,so 0*∞ is undefined.if it is defined,what the result will be?since 0 is pulling the result to 0 while ∞ pulling the result to ∞,who will win at last? no one knows. but there is a type of limit who is "0*∞" for example (x-1)/(x^2-4*x+3) where x approaches 1.
Hi,everybody,I am new here.
I have learned mathematics for many years and I admit I love them. I'm here to share my problems and pleasure in mathematics with you.
thanks,good luck for everyone.:P
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