lim x->oo (x^2)sin(1/x)
I did it for instead of .
In that case,
But we do not need to use that rule.
Instead for sufficiently large consider the reciprocal, i.e.
We can think of this reciprocal function as a composition where and . But at we define to be zero. Why? Because . This makes continous at zero.
Now and is continous at 0. By the limit composition rule the limit .
We have shown that . Which implies that its reciprocal is that is, .