functions

[Var1995]

pls help with this

[Dr. Strange]

I need a confirmation on this answer.

As n-> infinity , 1/n ->0

so near x=0 function becomes contant and is equal to 0
so f(0)=0 and f'(0)=0

[Var1995]

ur ans is correct

[Var1995]

 
i m solving for the domain.
i m getting -2≤|x| ≤6 but the answer includes only |x| ≤6. why are we ignoring the former part?

[Dr. Strange]

because mod x is a positive number and is always greater than -2

[Var1995]

could you tell me the values of x for the following:
1. |(x-3)/√9-x^2| ≤ 1
2. |3/ (4+sinx)| ≤ 1

[Dr. Strange]

1. x belongs to [0,3)
2. x can be anything

[Var1995]

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[Var1995]

someone pls help with this

(1+x^3) > 10^0 >1 this will give us (a) as the ans. are we using exponential function here? i m not getting it. someone pls explain.

[Dr. Strange]

both log functions have base 10

There will be two constraints: for inner log function condition is (1+x^3)>0 which gives x> -1

and outer log function gives constraint log (base 10) (1+x^3) >0
so  log (base 10) (1+x^3) > log (base 10) (1)
so 1+x^3 >0
so x>0

combining both constraints give us x belongs to (0,infinity)

[Var1995]

the domain of underoot cos(sin x) or underoot sin(cos x) is R. can u pls explain why that is so?

[Dr. Strange]

constraint is cos(sin x) >=0
   
However value of sinx varies from -1 to 1 which is always in 1st and 4th quadrant where cos is positive

so domain is R

[Dr. Strange]

However domain of sin(cos x) can't be R
X= pi,cos x =-1,sin cos x= sin -1= -0.84 which is not possible

In this case domain is cos x >=0
so x should be in 1st and 4th quadrant

[Tania123]

@Dr Strange For the first question, why wouldn't the answer be (a)

[Dr. Strange]

acc to ques, f(1)=0,f(0.5)=0,f(0.33)=0 and so on

As n increases,1/n decreases and so for higher values of n ,1/n tends to zero and function becomes continuous near 0 and is equal to 0.
However we can't say anything for f(3/4) or f(3/5) as they may or may not be zero.(since 4/3 or 5/3 is not an integer).
So f(x) is not 0 for  x belongs to (0,1)

[Var1995]

could you help me out with these problems
i m stuck in the last parts

[Dport]

I think it should be -3<x<3 for the first on. Correct me if I am doing it  wrong.

[Dr. Strange]

for first question,condition is
(log x)^2 -5*log x +6 >0 and x>0

(log x -2)(log x -3)>0
log x<2 or log x>3 and x>0
so 0<x<100 union x>1000



second question,
constraints 1. [x+ 0.5 ] is not equal to 0 or 1 where [ ] shows greatest integer
so minimum of greatest integer of x+0.5 =2 so minimum value of x =1.5

constraint 2. mod of (x ^ 2- x- 2) is not zero i.e  x is not 2 or -1

intersection of bth constraints shows x belongs to [3/2,infinity) -{2}

[Var1995]

thank u so much