True or false

Make use of L'Hospital Rule to determine which of the following statement is true or false:

1. 0×0=0

2. 0×∞=0

3. 0×∞=∞

4. ∞×∞=0

5. ∞×∞=∞

6. 0×0=∞

Solution for quesion 1&2:

Consider a rule:

0×c=0, where c is an indeterminate

Think whether c can be 0 or ∞

Consider an 0/0 scenario:

sin x / sin 2x when x approach 0

By L'Hospital rule, we have the limit:

cos x / 2 cos x = 1/2

Then we use the same rule to work on:

sin x / x^2 when x approach 0

Then we have cos x / 2x = 1/ 0 = ∞

Also consider x^2/ sin x when x approach 0:

We have 2x/ cos x = 0/1 =0

So c can be 0 or ∞

The statement 1&2 are then true.

Consider another indeterminate: ∞/∞

Basically using suitable limits,

statement 3, 5 are true.

4, 6 are false.

0/∞ = 0 × 0

∞/0 = ∞ × ∞

Straightforward