let f(x) be a real valued function defined for all x≥1, satisfying f(1) =1 and f'(x)=1/{x2 + (f(x))2} , then Lt f(x) :
x->∞
(a)does not exist (b)exists and less than ∩/4
(c) exists and less than 1+∩/4 (d) exists and equal to zero
donot jus prompt the answer and ask "is this the answer..please clarify"......i want the answer with the explanation!!!
24f'(x) > 0
=> f(x) is increasing
Now, f'(t)=1/(t2+f(t))
we know that for t>1 => f(t)>1
=>f'(t)=1/(t2+f(t)) <1/(t2+1)
Since f(1)=1 and F(x) is increaing fn.
x x
=> f(x)=1+∫f'(t) dt < 1 + ∫1/(1+t2) dt
1 1
∞
lim f(x) < 1+∫1/(1+t2) dt
x→∞ 1
=> lim f(x) < 1+(Ï€/4)
x→∞
=> (c)
19well done eureka.......now can you solve the problem on graphs also!!!!!