Please submit the following assignment before Feb 11, 11:59 PM to my email inbox, show all your work, comments and use word  symbols, no photos: 

lim𝑥2−7𝑥+12𝑥2−9𝑥−>3

                            lim𝑥−4−1(𝑥−5)𝑥−>5

     

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Certainly, let’s solve these limit problems step-by-step.

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### Problem 1: 
\[
\lim_{{x \to 3}} \frac{x^2 - 7x + 12}{x^2 - 9x}
\]

#### Step-by-Step Solution:
1. **Factor the Numerator and Denominator:**
   \[
   x^2 - 7x + 12 = (x - 3)(x - 4)
   \]
   \[
   x^2 - 9x = x(x - 9)
   \]

2. **Rewrite the Expression:**
   Substitute the factored forms:
   \[
   \frac{(x - 3)(x - 4)}{x(x - 9)}
   \]

3. **Evaluate the Limit by Substitution:**
   As \( x \to 3 \), substituting \( x = 3 \) directly would make the denominator zero, creating an indeterminate form. To address this, we simplify the expression further.

4. **Simplify and Attempt to Cancel Out Terms:**
   Unfortunately, no further simplification is possible here, so this limit doesn't cancel cleanly.

Please submit the following assignment before Feb 11, 11:59 PM to my email inbox, show all your work, comments and use word symbols, no photos:

lim𝑥2−7𝑥+12𝑥2−9𝑥−>3

lim𝑥−4−1(𝑥−5)𝑥−>5

Explore step-by-step solutions for limit problems involving factoring, including limits as x approaches specific constants. This example demonstrates algebraic techniques and limit laws essential for high school calculus. Perfect for Grades 11-12 students.

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