This helps to use the "extension of the real line" for this problem.

You have,

The first product exists (it is a standard limit with value of 1). The second does not exist but increases without bound. Thus, the answer is (c)

Let, , then, the limit of,2.find the value of limit lim x->inf x sin(1/x)

(a)0 (b)1 (c)2 (d)4 (e)none

can be found by evaluating the inner limit, .So the limit of,

by limit composition rule.

You can express,

The limit of the inner function exists and is zero because any polynomial overtakes a logarithm. Then by the limit composition rule.