W1.  lim

xS2

2x

2

-11x +14

x

2

-5x +6

It seems you're asking to find the limit of the following rational function as \( x \to 2 \):

\[
\lim_{x \to 2} \frac{2x^2 - 11x + 14}{x^2 - 5x + 6}
\]

Let's break it down and solve step by step:

### Step 1: Factorize the numerator and denominator
1. **Numerator:**
   The numerator is \( 2x^2 - 11x + 14 \). Let's factorize it:
   - Multiply the leading coefficient (2) by the constant (14) to get \( 2 \times 14 = 28 \).
   - We need two numbers that multiply to 28 and add to -11. These numbers are -7 and -4.

   So, the factorization of the numerator is:
   \[
   2x^2 - 11x + 14 = (2x - 7)(x - 2)
   \]

2. **Denominator:**
   The denominator is \( x^2 - 5x + 6 \). Factorizing this quadratic:
   - We need two numbers that multiply to 6 and add to -5. These numbers are -3 and -2.

   So, the factorization of the denominator is:
   \[
   x^2 - 5x + 6 = (x - 2)(x - 3)
   \]

### Step 2: Simplify the expression
Now that we have the factored form of both the numerator and denominator, the expression becomes:
\[
\frac{(2x - 7)(x - 2)}{(x - 2)(x - 3)}
\]

Cancel out the common factor \( (x - 2) \) from the numerator and denominator:
\[
\frac{2x - 7}{x - 3}
\]

### Step 3: Substitute \( x = 2 \) into the simplified expression
Now, substitute \( x = 2 \) into the simplified expression:
\[
\frac{2(2) - 7}{2 - 3} = \frac{4 - 7}{2 - 3} = \frac{-3}{-1} = 3
\]

### Final Answer:
\[
\lim_{x \to 2} \frac{2x^2 - 11x + 14}{x^2 - 5x + 6} = 3
\]

Let me know if you would like more details or have any questions!

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Here are 5 related questions you could explore:
1. How do you find limits when direct substitution results in an indeterminate form?
2. What is the difference between removable and non-removable discontinuities in a function?
3. Can you factor a higher-degree polynomial in a similar way?
4. How does the behavior of a function near a limit differ from its behavior at the limit?
5. What techniques can be used if a rational function can't be factored easily?

**Tip:** Always check if a limit leads to an indeterminate form, like \( \frac{0}{0} \), before applying algebraic simplifications.

lim x->2 (2x^2 - 11x + 14) / (x^2 - 5x + 6)

Explore how to solve the limit problem lim x->2 of (2x^2 - 11x + 14) / (x^2 - 5x + 6) by factoring and simplifying quadratic expressions. This step-by-step guide covers limit laws and factoring techniques for rational functions, suitable for Grades 10-12.

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