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The limit problem you've provided is as follows:

\[
\lim_{x \to 1^-} \left(\frac{1}{x+1}\right)\left(\frac{x+6}{x}\right)\left(\frac{3-x}{7}\right)
\]

To solve this limit as \( x \) approaches \( 1 \) from the left (\( x \to 1^- \)), we can evaluate the expression by directly substituting \( x = 1 \) into the function, considering the behavior from the left.

1. **Evaluate each component separately:**

   - \(\frac{1}{x+1}\) as \( x \) approaches 1 from the left:
     \[
     \frac{1}{x+1} = \frac{1}{1+1} = \frac{1}{2}
     \]

   - \(\frac{x+6}{x}\) as \( x \) approaches 1 from the left:
     \[
     \frac{x+6}{x} = \frac{1+6}{1} = \frac{7}{1} = 7
     \]

   - \(\frac{3-x}{7}\) as \( x \) approaches 1 from the left:
     \[
     \frac{3-x}{7} = \frac{3-1}{7} = \frac{2}{7}
     \]

2. **Multiply the results:**
   \[
   \lim_{x \to 1^-} \left(\frac{1}{x+1}\right)\left(\frac{x+6}{x}\right)\left(\frac{3-x}{7}\right) = \frac{1}{2} \times 7 \times \frac{2}{7} = \frac{1 \times 7 \times 2}{2 \times 7} = \frac{14}{14} = 1
   \]

Therefore, the limit is:

\[
\lim_{x \to 1^-} \left(\frac{1}{x+1}\right)\left(\frac{x+6}{x}\right)\left(\frac{3-x}{7}\right) = 1
\]

Would you like more details on this solution or have any questions?

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Here are five questions related to this problem:
1. What happens if we evaluate the limit as \( x \) approaches 1 from the right instead?
2. How would the result change if any of the fractions had a discontinuity at \( x = 1 \)?
3. Can this limit be evaluated using L'Hôpital's Rule if it resulted in an indeterminate form?
4. What is the significance of the direction \( x \to 1^- \) in this context?
5. How would you approach this problem if one of the terms was undefined at \( x = 1 \)?

**Tip:** Always check if the limit results in an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), as that might indicate the need for further techniques like L'Hôpital's Rule or algebraic simplification.

Find the limit as x approaches 1 from the left for the expression: (1/(x+1)) * ((x+6)/x) * ((3-x)/7).

Explore the solution to the limit problem as x approaches 1 from the left for the expression (1/(x+1)) * ((x+6)/x) * ((3-x)/7). The solution covers key concepts in limits and algebra, and is suitable for students in Grades 11-12.

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