The given expression to simplify is:

$\lim_{t \to 1} \frac{nt^{n+1} - (n+1)t^n + 1}{(t-1)^2}$

To solve this limit, we will approach the problem step by step.

Step 1: Recognizing the Form

At $t = 1$, direct substitution into the expression would give us an indeterminate form of $\frac{0}{0}$. This suggests that we need to apply techniques like L'Hopital's rule or series expansion to evaluate the limit.

Step 2: Using L'Hopital's Rule

Since we have an indeterminate form $\frac{0}{0}$, we can differentiate the numerator and denominator separately and then evaluate the limit.

The numerator is:

$f(t) = nt^{n+1} - (n+1)t^n + 1$

The denominator is:

$g(t) = (t-1)^2$

Step 3: Differentiating the Numerator and Denominator

First, differentiate the numerator and denominator.

For the numerator $f(t)$: