<s>[INST] Find $\lim_{x \to 3} \frac{x^{15} - 3^{15} - 15 \cdot 3^{14}(x-3)}{(x-3)^2}$ Please help me find:

$$
\lim_{x \to 3} \frac{x^{15} - 3^{15} - 15 \cdot 3^{14}(x-3)}{(x-3)^2}
$$

I cannot use L'Hospital's rule.
I tried to eliminate $x-3$, but I have no idea what to do next.
$$
\lim_{x \to 3} \frac{x^{15} - 3^{15} - 15 \cdot 3^{14}(x-3)}{(x-3)^2} = 
\lim_{x \to 3} \frac{x^{14} + 3x^{13} + ... + 3^{14} - 15 \cdot 3^{14}}{x-3}
$$
 [/INST] Now, $$\frac{x^{14}+3x^{13}+...+3^{13}x-14\cdot3^{14}}{x-3}=\frac{x^{14}-3^{14}}{x-3}+3\cdot\frac{x^{13}-3^{13}}{x-3}+...+3^{13}\rightarrow$$
$$=14\cdot3^{13}+3\cdot13\cdot3^{12}+...+3^{13}=(14+13+...+1)3^{13}=105\cdot3^{13}=35\cdot3^{14}.$$
I used the following:
$$a^n-b^n=(a-b)\left(a^{n-1}+a^{n-2}b+...+ab^{n-2}+b^{n-1}\right)$$  for natural $n\geq2$.
We see that in the big  brackets we have $n$ terms $a^n$ for $a=b$.
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