You could claim the same thing about the original limit. Both the numerator and denominator in that case were zero, so the limit doesn't exist. So why bother doing anything? Yet you found by plotting the function that the answer in the book is correct. Also, your claim also applies to the example I gave. Again, both the numerator and denominator are equal to zero when you set x=0, so you claim the limit doesn't exist. Yet the limit exists and is equal to 1.The denominator is zero here as well. User vela, by my logic, the limit doesn't exist for that function because it is undefined due to both denominator and numerator being equal to zero.
In evaluating limits, 0/0 is what's called an indeterminate form. You can't tell what the limit is equal to or if it even exists without doing a bit more work. Halls told you the technique to use in this particular case so you can evaluate the limit.