lim x-->a [f(x) - g(x)] = lim x-->a f(x) - lim x-->a g(x)?

Question

This is a practice problem with the answer in the back of the text; the author claims it is a *false* statement. My question is, 'Why is it a false statement?'

lim x-->4 [ (( 2x ) / (x-4) ) - ( 8 / (x-4)) ] = lim x-->4 [( 2x ) / (x-4)] - lim x-->4 [( 8 / (x-4) ]

One of the limit laws states that:
lim x-->a [f(x) - g(x)] = lim x-->a f(x) - lim x-->a g(x)

Answer 1

The law applies only when the limits of f and g are defined. In your example, the limit of f - g is defined, but those of f and g are not. As x -> 4+, both go to infinity; as x -> 4-, both go to minus infinity. But the two-sided limit is not defined, while the limit of the difference is.

Answer 2

The back of the book is wrong. The limit of a difference is equal to the difference of the limits.

Answer 3

The book is right on this one. If you solve the left side of the equation it equals 2. When you solve the right side of the equation it is infinity - infinity. So (2 = infinity - infinity) is not true.