Continuity Problem

Calculus Level 3

Let \(f(x)\) be a piecewise function such that: \[ \large f(x) = \begin{cases} \begin{align} & \cfrac{x^{2^{32}} - (4^{16})x + 4^{16} - 1}{(x - 1)^2} & , \hspace{1cm} x \ne 1 \\ & k & , \hspace{1cm} x = 1 \end{align} \end{cases} \]

Let \(k = a^b \times (a^c - 1) \), where the value of \(k\) make \(f(x)\) continuous at 1.

If \(a \) is a prime number and \(b < c\) , solve for \(a + b + c\) .

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