Math Snap

STEP 1

What is this asking?
What happens to the fraction $\frac{x}{2-x}$ as $x$ gets really close to $2$ from the left?
Watch out!
We're approaching $2$ from the left, meaning $x$ is slightly less than $2$, not greater!

STEP 2

1. Analyze the numerator
2. Analyze the denominator
3. Combine the analysis

STEP 3

As $x$ approaches $2$ from the left, the numerator, $x$, gets closer and closer to $\textbf{2}$.
It's like walking towards a finish line – you're getting closer and closer to your destination!

STEP 4

Since we're approaching $2$ from the left, $x$ is a little less than $2$.
Think of it as $x = 2 - \text{tiny bit}$.

STEP 5

So, the denominator, $2 - x$, becomes $2 - (2 - \text{tiny bit})$.

STEP 6

This simplifies to $\textbf{tiny bit}$.
The denominator is a small positive number getting closer and closer to zero!

STEP 7

We have a numerator approaching $\textbf{2}$ and a denominator, a tiny positive number, approaching $\textbf{0}$.
Imagine dividing $2$ by a super tiny number, like $0.000001$.
The result is a huge number!

STEP 8

As the "tiny bit" gets even tinier, the fraction $\frac{2}{\text{tiny bit}}$ gets larger and larger.
It grows without bound!

STEP 9

Therefore, the limit is positive infinity.

$$\lim_{x \rightarrow 2^{-}} \frac{x}{2-x} = \infty$$