Question

The following statement is FALSE: If lim x!6 [f(x)g(x)] exists, then the limit must be f(6)g(6).

Give an example of two functions f(x) and g(x) that demonstrate the falsity of this state- ment - that is, two functions f(x) and g(x) such that lim x!6 [f(x)g(x)] exists, but is not equal to f(6)g(6). Explain your answer.

Answer 1

we are given

exists

then limit must be

this is FALSE

Example:

Let's assume

Firstly, we can find limit

we can cancel it

now, we can find f(6) and g(6)

so,

...which is indeterminant

So, we can see that limit value is 1

and this value is indeterminant

so, they are not equal

Hence, this is FALSE......Answer