Question

How do you find `lim (1-x)/(2-x)` as `x->2`?

Answer 1

The limit does not exist.

Explanation:

As `xrarr2`, the numerator goes to `-1` and the denominator goes to `0`. The limits does not exist.

We can say more about why the limit does not exist.

As `x` approaches `2` from the right (through values greater than `2`), `2-x` is a negative number close to `0` (a negative fraction).
So, as `xrarr2^+`, the quotient `(1-x)/(2-x)` increases without bound.
We write `lim_(xrarr2^+)(1-x)/(2-x) = oo`

As `x` approaches `2` from the left, `2-x` is a positive number close to `0`.
So, as `xrarr2^-`, the quotient `(1-x)/(2-x)` decreases without bound.
We write `lim_(xrarr2^-)(1-x)/(2-x) = -oo`