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To determine the value of \( a \) so that the given function is continuous at \( x = 1 \), we need to ensure that the limit of \( f(x) \) as \( x \) approaches 1 from both sides equals \( f(1) = a \).

The function \( f(x) \) is given as:

\[
f(x) = \begin{cases} 
\frac{1 - x^3}{1 - x^2}, & x \neq 1 \\
a, & x = 1 
\end{cases}
\]

### Step 1: Calculate the limit of \( f(x) \) as \( x \to 1 \).

The expression for \( x \neq 1 \) is \( \frac{1 - x^3}{1 - x^2} \). To compute the limit as \( x \to 1 \), we can first simplify this expression.

\[
1 - x^3 = (1 - x)(1 + x + x^2)
\]
\[
1 - x^2 = (1 - x)(1 + x)
\]

Thus, the function becomes:

\[
f(x) = \frac{(1 - x)(1 + x + x^2)}{(1 - x)(1 + x)}, \quad x \neq 1
\]

We can cancel out the \( 1 - x \) term (since \( x \neq 1 \)):

\[
f(x) = \frac{1 + x + x^2}{1 + x}, \quad x \neq 1
\]

### Step 2: Take the limit as \( x \to 1 \).

Now we compute:

\[
\lim_{x \to 1} \frac{1 + x + x^2}{1 + x} = \frac{1 + 1 + 1^2}{1 + 1} = \frac{1 + 1 + 1}{1 + 1} = \frac{3}{2}
\]

### Step 3: Set \( a \) to ensure continuity.

For the function to be continuous at \( x = 1 \), we need:

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

Thus, we set:

\[
a = \frac{3}{2}
\]

### Final Answer:
The value of \( a \) must be \( \frac{3}{2} \) for the function to be continuous at \( x = 1 \).

---

Would you like more details or have any questions? Here are five related questions to explore further:

1. How do you define continuity for a piecewise function?
2. What is the significance of factoring and simplifying expressions in limits?
3. How do you handle indeterminate forms like \( \frac{0}{0} \) when computing limits?
4. What is the difference between continuity and differentiability at a point?
5. Can you explain the concept of a removable discontinuity?

**Tip:** When simplifying limits, always check for common factors that may cancel out indeterminate forms.

Determinar el valor de a de manera que la función sea continua.
 f(x) = { (1 - x^3) / (1 - x^2), x ≠ 1 ; a, x = 1 }

Learn how to find the value of a for the piecewise function to be continuous at x = 1. This solution covers key concepts in limits, factoring polynomials, and continuity, with step-by-step calculations.

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