Evaluate the limit: lim (t->0) (10^t - 2^t - 5^t + 1) / (t * tan(t))

Okay, let's tackle this limit problem.

Step 1: Check for indeterminate form.
Substitute $t=0$ into the expression:
Numerator: $10^0 - 2^0 - 5^0 + 1 = 1 - 1 - 1 + 1 = 0$.
Denominator: $0 \cdot \tan(0) = 0 \cdot 0 = 0$.
This is an indeterminate form of type $\frac{0}{0}$, so we can apply L'Hôpital's Rule.

Step 2: Apply L'Hôpital's Rule (first time).
Take the derivative of the numerator and the denominator:

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