Apply the definition of continuity to identify any points of discontinuity in the function f(x) = 
(𝑥−2)/(𝑥^2−3𝑥+2)

To determine the points of discontinuity in the function \( f(x) = \frac{x - 2}{x^2 - 3x + 2} \), we can use the definition of continuity, which states that a function is continuous at a point \(x = a\) if the following three conditions hold:

1. \(f(a)\) is defined.
2. \(\lim_{x \to a} f(x)\) exists.
3. \(\lim_{x \to a} f(x) = f(a)\).

The first step is to factor the denominator \(x^2 - 3x + 2\).

### Step 1: Factor the denominator
The quadratic in the denominator can be factored as:
\[
x^2 - 3x + 2 = (x - 1)(x - 2).
\]

So the function becomes:
\[
f(x) = \frac{x - 2}{(x - 1)(x - 2)}.
\]

### Step 2: Simplify the function
We notice that the numerator \(x - 2\) and one factor in the denominator, \(x - 2\), are the same. This means we can cancel the common factor, but we must note that this cancellation is only valid for \(x \neq 2\). So the simplified function is:
\[
f(x) = \frac{1}{x - 1}, \quad \text{for} \quad x \neq 2.
\]

### Step 3: Identify points of discontinuity
Now we analyze the points where the function is not continuous:

- **At \(x = 2\)**: The original function is undefined because the denominator becomes zero when \(x = 2\). Even after simplification, the function still has no definition at \(x = 2\). Thus, \(x = 2\) is a **removable discontinuity**, because if we redefine the function at this point, continuity could be restored.
  
- **At \(x = 1\)**: The simplified function \(f(x) = \frac{1}{x - 1}\) is undefined at \(x = 1\) because the denominator becomes zero. This creates a **non-removable (infinite) discontinuity** at \(x = 1\), as the limit does not exist at this point.

### Conclusion
The function has two discontinuities:

1. A **removable discontinuity** at \(x = 2\).
2. A **non-removable discontinuity** at \(x = 1\).

Would you like further details or have any questions? 

Here are some related questions to further explore this topic:

1. What is the difference between removable and non-removable discontinuities?
2. How can we redefine the function to remove the discontinuity at \(x = 2\)?
3. How do you determine the limits at points of discontinuity?
4. Can a function have both types of discontinuities at the same point?
5. How would the graph of this function look around \(x = 1\) and \(x = 2\)?

**Tip**: To handle rational functions like this one, always check for factors that cancel out in the numerator and denominator, as they often indicate removable discontinuities.

Apply the definition of continuity to identify any points of discontinuity in the function f(x) = (x−2)/(x^2−3x+2)

Explore how to identify points of discontinuity in the rational function f(x) = (x−2)/(x^2−3x+2) using the definition of continuity. The solution involves factoring the quadratic equation and determining removable and non-removable discontinuities. Suitable for students in Grades 11-12.

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