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The problem requires finding the two-sided limit \(\lim_{x \to 1} f(x)\), given the piecewise function:

\[
f(x) =
\begin{cases}
-2x + 3 & \text{if } x \leq 1, \\
-7x + 8 & \text{if } x > 1.
\end{cases}
\]

### Step 1: Evaluate the Left-Hand Limit (\(\lim_{x \to 1^-} f(x)\))
For \(x \leq 1\), \(f(x) = -2x + 3\). Substituting \(x = 1\) into \(-2x + 3\):

\[
\lim_{x \to 1^-} f(x) = -2(1) + 3 = -2 + 3 = 1.
\]

### Step 2: Evaluate the Right-Hand Limit (\(\lim_{x \to 1^+} f(x)\))
For \(x > 1\), \(f(x) = -7x + 8\). Substituting \(x = 1\) into \(-7x + 8\):

\[
\lim_{x \to 1^+} f(x) = -7(1) + 8 = -7 + 8 = 1.
\]

### Step 3: Compare the Left-Hand and Right-Hand Limits
Since \(\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = 1\), the two-sided limit exists, and its value is:

\[
\lim_{x \to 1} f(x) = 1.
\]

### Final Answer:
**A. 1**

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Would you like further details or have any questions?  

### Expanding Questions:
1. What happens if the left-hand and right-hand limits differ at a point?
2. Can you describe the conditions for a limit to exist?
3. How would this problem change if \(x = 1\) was excluded from the function definition?
4. Why is continuity not necessary for limits to exist?
5. How can we visualize this piecewise function graphically?

### Tip:
When working with piecewise functions, always evaluate the limit from both sides of the critical point to ensure consistency.

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. Let f(x) = { -2x + 3 if x ≤ 1; -7x + 8 if x > 1 }. Find lim_{x → 1} f(x).

Discover how to find the two-sided limit lim_{x → 1} f(x) for a piecewise function. Learn the steps to evaluate left-hand and right-hand limits and verify their equality. Ideal for Grades 11-12 students studying calculus.

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