A) \[\frac{{{a}^{2}}}{2}{{(\alpha -\beta )}^{2}}\]
B) \[0\]
C) \[\frac{-{{a}^{2}}}{2}{{(\alpha -\beta )}^{2}}\]
D) \[\frac{1}{2}{{(\alpha -\beta )}^{2}}\]
Correct Answer: A
Solution :
\[\underset{x\to \alpha }{\mathop{\lim }}\,\frac{1-\cos (a{{x}^{2}}+bx+c)}{{{(x-\alpha )}^{2}}}\] \[=\underset{x\to \alpha }{\mathop{\lim }}\,\frac{2{{\sin }^{2}}\left( \frac{a{{x}^{2}}+bx+c}{2} \right)}{{{(x-\alpha )}^{2}}}\] \[=\underset{x\to \alpha }{\mathop{\lim }}\,\frac{2{{\sin }^{2}}\left( \frac{a}{2}(x-\alpha )(x-\beta ) \right)}{{{\left( \frac{a}{2} \right)}^{2}}{{(x-\alpha )}^{2}}{{(x-\beta )}^{2}}}\] \[=\underset{x\to \alpha }{\mathop{\lim }}\,\frac{{{a}^{2}}}{2}{{(x-\beta )}^{2}}=\frac{{{a}^{2}}}{2}{{(\alpha -\beta )}^{2}}\]
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