To solve the limit

$\lim_{x \to 1} \frac{x - x^4}{1 - x},$

we need to simplify the expression to determine what should go into the first box. Let's break it down:

1. Factor the numerator: The numerator is $x - x^4$, and we can factor it as:

   $x - x^4 = x(1 - x^3) = x(1 - (x)(x^2)) = x(1 - x)(1 + x + x^2).$

2. Simplify the expression: Now we can substitute this factored form back into the original expression:

   $\frac{x(1 - x)(1 + x + x^2)}{1 - x}.$

   The $1 - x$ terms cancel out, leaving us with:

   $\lim_{x \to 1} x(1 + x + x^2).$

3. Evaluate the limit: We now evaluate the limit as $x \to 1$:

   $1(1 + 1 + 1^2) = 1(1 + 1 + 1) = 1 \times 3 = 3.$

Therefore, the first box should contain $x(1 + x + x^2)$, and the final answer for the limit is $3$.

Let me know if you'd like further clarification or additional details!

Relative Questions:

1. What are common techniques to simplify limits involving polynomials?
2. How do you factor higher-degree polynomials, such as $x^4$?
3. What role does the cancellation of terms play in evaluating limits?
4. How can you verify that a limit results in a removable discontinuity?
5. What is the importance of limits in calculus?

Tip: When you have an indeterminate form in limits (like $\frac{0}{0}$), try factoring or applying L'Hopital's Rule.