Question

# Let $$\alpha$$ and $$\beta$$ be the roots of $$\mathrm{a}\mathrm{x}^{2}+\mathrm{b}\mathrm{x}+\mathrm{c}= 0$$, then $$\displaystyle \lim_{x\displaystyle \rightarrow\alpha}\displaystyle \frac{1-\cos(ax^{2}+bx+c)}{(x-\alpha)^{2}}=$$

A
a2(αβ)22
B
a22(αβ)2
C
a2(αβ)2
D
a22(αβ)2

Solution

## The correct option is B $$\displaystyle \frac{a^{2}(\alpha-\beta)^{2}}{2}$$$$\displaystyle \lim_{x\rightarrow \alpha }\frac{1-\cos \left ( ax^{2}+bx+c \right )}{\left ( x-\alpha \right )^{2}}=\lim_{x\rightarrow \alpha }\frac{2\sin ^{2}\left ( \frac{ax^{2}+bx+c}{2} \right )}{\left ( x-\alpha \right )^{2}}$$$$\displaystyle =\lim_{x\rightarrow \alpha }\frac{4\sin ^{2}\left [ \frac{a\left ( x-\alpha \right )\left ( x-\beta \right )}{2} \right ]}{a^{2}\left ( x-\alpha \right )^{2}\left ( x-\beta \right )^{2}}\times \frac{a^{2}\left ( x-\beta \right )^{2}}{2}=\frac{a^{2}}{2}\left ( \alpha -\beta \right )^{2}$$Mathematics

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