Question

Find each of the limits without a calculator. Use the limits as an aide to sketch the graph of f(x)= (x-2)/(|x|-2) for-10<_x<_10 and -6<_y<_6. Be sure to clearly indicate any asymptotes, holes or other important characteristics.

lim x->∞ f(x)

Answer 1

recall the definition of |u|. It is
u for u>=0, and -u for u<0. So, your function
f(x) =
(x-2)/(x-2) for x >= 2
(x-2)/(-x-2) for x < 2

Now you can easily find the necessary limits. Then, peek at the graph below to verify your calculations.

Answer 2

Would I do the same thing as x approaches:

Infinity
-2+
-2-
2+
2-