We are solving the limit of the expression:

$\lim_{x \to 1} \frac{x^n + x^{n-1} + \dots + x - n}{x - 1}.$

Here is the step-by-step solution without using derivatives:

Step 1: Recognize the structure of the numerator

The numerator is a sum of powers of $x$ from $x^n$ down to $x^1$, with $n$ terms. Subtracting $n$ at the end means the numerator is:

$x^n + x^{n-1} + \cdots + x - n.$

This expression can be rewritten as: $\sum_{k=1}^n x^k - n.$

Step 2: Factorize using the formula for the sum of powers

The sum $x^n + x^{n-1} + \cdots + x^1$ can be rewritten using the formula for the sum of a geometric series:

$x^n + x^{n-1} + \cdots + x = x \cdot \frac{x^n - 1}{x - 1}.$

Thus, the numerator becomes: $x \cdot \frac{x^n - 1}{x - 1} - n.$

Step 3: Simplify the fraction

Now the full expression becomes: $\lim_{x \to 1} \frac{x \cdot \frac{x^n - 1}{x - 1} - n}{x - 1}.$

Step 4: Handle the indeterminate form

As $x \to 1$, the denominator $x - 1 \to 0$, and the numerator also tends to 0. To resolve this, we rewrite the numerator and evaluate the limit.

Step 5: Expand $x^n - 1$ using the binomial theorem

The term $x^n - 1$ can be expanded as: $x^n - 1 = (x - 1)(x^{n-1} + x^{n-2} + \cdots + 1).$

Substituting this back into the expression: $x \cdot \frac{x^n - 1}{x - 1} = x \cdot (x^{n-1} + x^{n-2} + \cdots + 1).$

Thus, the numerator becomes: $x \cdot (x^{n-1} + x^{n-2} + \cdots + 1) - n.$

At $x = 1$, the term $x^{n-1} + x^{n-2} + \cdots + 1$ has exactly $n$ terms, each equal to 1. So this sum equals $n$, and: $\text{Numerator at } x = 1 = n - n = 0.$

Step 6: Use the expansion for small $x - 1$

Now expand $x^{n-1}, x^{n-2}, \dots$ in terms of $x - 1$. For small $x$, substitute: $x^k = 1 + k(x - 1) + \mathcal{O}((x - 1)^2).$

The numerator becomes: $1 + 2(x - 1) + 3(x - 1) + \dots + n(x - 1) - n.$

Summing the coefficients of $x - 1$: $\text{Coefficient of } (x - 1) = 1 + 2 + \cdots + n = \frac{n(n+1)}{2}.$

Thus, as $x \to 1$, the numerator divided by $x - 1$ becomes: $\frac{n(n+1)}{2}.$

Final Answer:

$\lim_{x \to 1} \frac{x^n + x^{n-1} + \cdots + x - n}{x - 1} = \frac{n(n+1)}{2}.$

Let me know if you have questions about the steps or want further explanation! Here are related questions:

1. What is the significance of the sum formula used here?
2. How does the binomial expansion simplify higher powers of $x$?
3. Can we use a derivative method to verify this result?
4. What happens if $n = 1$ or $n = 2$? Test those cases.
5. How does this problem relate to the concept of a geometric series?

Tip: Always simplify terms systematically to avoid handling too many variables at once!