Mathematics - Limits Question with Solution | TestHub

$$\begin{gathered} \underset{x\to \beta }{\lim }\frac{e^{2\left(x^2+bx+c\right)}-1-2\left(x^2+bx+c\right)}{{\left(x-\beta \right)}^2} \\ \underset{x\to \beta }{\lim }\frac{1\left(1+{\displaystyle \frac{2\left(x^2+bx+c\right)}{1!}}+{\displaystyle \frac{2^2{\left(x^2+bx+c\right)}^2}{2!}}+...\right)-1-2\left(x^2+bx+c\right)}{{\left(x-\beta \right)}^2} \\ \underset{x\to \beta }{\lim }\frac{2{\left(x^2+bx+c\right)}^2}{{\left(x-\beta \right)}^2} \\ \underset{x\to \beta }{\lim }\frac{2{\left(x-\alpha \right)}^2{\left(x-\beta \right)}^2}{{\left(x-\beta \right)}^2} \\ 2{\left(\beta -\alpha \right)}^2=2\left(b^2-4c\right) \end{gathered}$$