Math Help - Double-check this limit please

Can someone double-check my work here? Thanks, if you can.

$
\lim_{x\to\infty}[x(1-cos(1/x))]
$

Put x to the denominator to use L'Hopital's rule, etc. Applied chain rules, etc.

$
\lim_{x\to\infty}[sin(1/x)]
$

Thus, if x goes to infinity, the above is sin(0) = 0. Final Answer: Zero

2. Yes, it's okay, but you don't need that rule, you can turn that limit into a known one with a simple substitution.

3. Oh, really? What is this substitution?

4. $t=\frac1x.$

5. tx=1

so itequals to lim [(1-cost)/t]
t->0