if f(x)= sin-1([x]+x)/[x] ,[x]≠0
= 0 ,[x]=0
then find lim f(x)
x→0
106manish sir: how f(0+) = 0? it is undefined for 1>x≥0 isnt it?
ive a doubt. Take this case here the function has domain (-∞,0)U[1,∞)
So here lim(x→0) will exist or not??
In many cases ive seen that the limit is taken as f(0-) in this case as the function itself is not defined at RHL(as we do while checking if a function is continuous). While in other cases the answer is given as lim does not exist bcz RHL is not defined
1357Here I also initially thought the same
but here f(x)=0 for [x]=0 that means x in [0,1)
106oh yeah!!! i thot it was x=0.
anyways what abt the dbt in my last para? wud we say limit does not exist? if it was given x=0 instead of [x]=0 ??
106the value wud not have been defined. but im asking abt limit. I guess that since we had discussed this and got no satisfactory conclusion we shud wait for others... please wait abhisek b4 answering
1LHL=\lim_{h\rightarrow 0}f(0-h)=\frac{sin^{-1}([-h]-h)}{-[h]}=\frac{sin^{-1}(-1-h)}{-1}=\frac{\pi }{2}
RHL=\lim_{h\rightarrow 0}f(0+h)=0
