If $\displaystyle \alpha $ and $\displaystyle \beta$ are the roots of the equation $\displaystyle ax^{2}+bx+c= 0$, then the value of $\displaystyle \underset{x\rightarrow a}{lim}\left [ ax^{2}+bx+c+1 \right ]^{2/\left ( x-a \right )}$ is:

Substituting x = a into the expression ax^2 + bx + c + 1 yields a^3 + ab + c + 1 = 1. This creates a 1^infinity indeterminate form for the limit. Using the standard limit identity lim (x->0) (1 + x)^(1/x) = e, we rewrite the expression as e^(lim (x->a) 2*(ax^2 + bx + c) / (x - a)). Because alpha is a root of ax^2 + bx + c, we can factor the numerator as a*(x - alpha)(x - beta), making the limit e^(lim (x->a) 2*a(x - alpha)(x - beta) / (x - a)). Assuming a is related to alpha such that a = alpha, substituting x = alpha gives the limit e^(2*a(alpha - beta)).

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