Limit of a GIF function

Question

The !!lim_(x->0)[(sinx)/x]!! is:
a) !!1!!
b) !!-1!!
c) !!0!!
d) None
Here !![ ]!! is greatest integer function.

Solution

Many students have made the mistake of assuming:

!!lim_(x->0)[(sinx)/x]=[lim_(x->0)(sinx)/x]!!, but this does not hold.

We find the limit of !![(sinx)/x]!! as !!x->0^+!! and !!x->0^-!!

Consider the function !!f(x)=sinx-x!!

Now !!f'(x)=cosx-1!!, !!f'(x)<0 forall x!=2npi!!.

Since !!f(x)=0!!, for !!x>0!! we have !!f(x)<0!! and !!f(x)>0!! for !!x<0!!

This implies

!!sinx< x !! for !!x>0!!, !!sinx< x!! for !!x<0!!

!!(sinx)/x<1!! for !!x!=0!!

Some of us analyzed the !!x<0!! case incorrectly

So we get that !![(sinx)/x]=0 "for" x!=0!!.

Hence we get !!lim_(x->0)[(sinx)/x]=0!!

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