how do you find the limit of the function f\left(x\right)=\frac{x^{2}-3x+2}{x^{2}+x-6}

To find the limit of the function \( f(x) = \frac{x^2 - 3x + 2}{x^2 + x - 6} \), you need to analyze the function's behavior as \( x \) approaches a specific value. 

Let's break it down step by step.

### Step 1: Factor the numerator and denominator
First, factor both the numerator and the denominator of the function.

- The numerator is \( x^2 - 3x + 2 \). This can be factored as:
  \[
  x^2 - 3x + 2 = (x - 1)(x - 2)
  \]
- The denominator is \( x^2 + x - 6 \). This can be factored as:
  \[
  x^2 + x - 6 = (x - 2)(x + 3)
  \]

So, the function can be rewritten as:
\[
f(x) = \frac{(x - 1)(x - 2)}{(x - 2)(x + 3)}
\]

### Step 2: Simplify the expression
Notice that the factor \( (x - 2) \) appears in both the numerator and the denominator. Assuming \( x \neq 2 \), these terms can be canceled out:
\[
f(x) = \frac{x - 1}{x + 3}
\]
However, it's important to remember that the original function is undefined at \( x = 2 \) because the denominator becomes zero there.

### Step 3: Find the limit
Now, you can find the limit as \( x \) approaches a specific value. Let's consider two common cases:

#### Case 1: \( \lim_{x \to 2} f(x) \)
Since the function is undefined at \( x = 2 \), we instead find the limit of the simplified expression:
\[
\lim_{x \to 2} f(x) = \lim_{x \to 2} \frac{x - 1}{x + 3} = \frac{2 - 1}{2 + 3} = \frac{1}{5}
\]

#### Case 2: \( \lim_{x \to \infty} f(x) \)
For very large values of \( x \), the behavior of the function is dominated by the highest degree terms in the numerator and the denominator. The highest degree terms are \( x \) in the numerator and \( x \) in the denominator:
\[
\lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{x - 1}{x + 3} = \lim_{x \to \infty} \frac{x(1 - \frac{1}{x})}{x(1 + \frac{3}{x})} = \frac{1}{1} = 1
\]

### Summary of Limits
- \( \lim_{x \to 2} f(x) = \frac{1}{5} \)
- \( \lim_{x \to \infty} f(x) = 1 \)

Would you like details on any of these steps, or do you have any other questions?

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Here are 5 related questions to explore:

1. How do you handle limits when the function is undefined at a certain point?
2. What is the importance of factoring in finding limits?
3. How do you approach limits that involve infinity?
4. What is a removable discontinuity in the context of limits?
5. How can limits be used to find the derivative of a function?

**Tip:** Always consider simplifying the function before applying limit rules, especially if the function is undefined at a point. This can often reveal removable discontinuities.

Learn how to find the limit of the function f(x) = (x^2 - 3x + 2) / (x^2 + x - 6) using factorization and simplification. Understand the behavior of rational functions and their limits. Suitable for advanced high school students and college calculus students.

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