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

##### 1 Answer
Dec 16, 2016

The limit does not exist.

#### Explanation:

As $x \rightarrow 2$, 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 $x \rightarrow {2}^{+}$, the quotient $\frac{1 - x}{2 - x}$ increases without bound.
We write ${\lim}_{x \rightarrow {2}^{+}} \frac{1 - x}{2 - x} = \infty$

As $x$ approaches $2$ from the left, $2 - x$ is a positive number close to $0$.
So, as $x \rightarrow {2}^{-}$, the quotient $\frac{1 - x}{2 - x}$ decreases without bound.
We write ${\lim}_{x \rightarrow {2}^{-}} \frac{1 - x}{2 - x} = - \infty$