Question

How do you evaluate `\lim _ { x \rightarrow - \infty } \frac { | x - 4| } { 2x }`?

Answer 1

`-1/2`

Explanation:

`abs(x-4)/(2x) = abs(x/absx-4/absx)/(2 x/absx)`

now `lim_(x->-oo) f(x) = lim_(y->oo)f(-y)` so

`lim_(x->oo) abs(x/absx-4/absx)/(2 x/absx) = lim_(y->oo) abs(-y/absy-4/absy)/(-2 y/absy) = abs(-1 -lim_(y->oo)4/abs y)/(-2) = abs(-1)/(-2) = -1/2`