To solve the limit

$\lim_{x \to a} \frac{(x - a)^{42} - 7x + 7a}{x - a},$

we can approach the problem as follows:

Step 1: Factor and Simplify

Firstly, observe that direct substitution $x = a$ into the limit results in an indeterminate form $\frac{0}{0}$. So, we need to simplify the expression.

We can separate the terms in the numerator:

$(x - a)^{42} - 7x + 7a = (x - a)^{42} - 7(x - a).$

This allows us to rewrite the limit as:

$\lim_{x \to a} \frac{(x - a)^{42} - 7(x - a)}{x - a}.$

Step 2: Factor the Numerator

The numerator can now be factored:

$(x - a)^{42} - 7(x - a) = (x - a)\left[(x - a)^{41} - 7\right].$

Step 3: Simplify the Expression

Now, we can cancel the common factor of $(x - a)$ in the numerator and the denominator:

[ \lim_{x \to a} \frac{(x - a)\left[(x - a)^{41} - 7\right]}{x - a} = \lim_{x \to a} \left[(x - a)^{41} - 7\right]. ]

Step 4: Apply the Limit

Finally, substitute $x = a$ into the simplified expression:

$\lim_{x \to a} \left[(x - a)^{41} - 7\right] = (a - a)^{41} - 7 = 0 - 7 = -7.$

Final Answer:

The limit is

$\boxed{-7}.$

Would you like more details on any of the steps or have further questions?

Here are some related questions you might find interesting:

1. How do you approach limits that result in an indeterminate form?
2. What are other methods for simplifying limits, such as L'Hôpital's Rule?
3. Can you explain the importance of factoring in solving limit problems?
4. How would the solution change if the power in the expression was different, say $(x-a)^{n}$ instead of $(x-a)^{42}$?
5. What is the role of continuity in evaluating limits?

Tip: When you encounter a limit problem that results in an indeterminate form, look for common factors to cancel out or consider alternative methods such as L'Hôpital's Rule for more complex cases.