Limit of x*sin(1/x) as X approaches 0

Question

After those steps, the result is wrong, and I can't understand what's wrong.

lim(x→0) x·sin(1/x) =

a = 1/x

lim(x→0) x·sin(a)·a/a=

lim(x→0) (x·sin(a)·a)/a=

since lim(x→0) sin(a)/a = 1, than:

lim(x→0) x·a=

lim(x→0) x·1/x=

lim(x→0) x/x = 1

But the right answer is 0, how can this be possible?

Answer 1

since sine is always between -1 and 1, consider the following inequality:

lim_(x→0) x(-1) ≤ lim_(x→0) x sin 1/x ≤ lim_(x→0) x(1)

lim_(x→0) -x ≤ lim_(x→0) x sin 1/x ≤ lim_(x→0) x

use the squeeze theorem to show the limit in the middle is 0 too