A) \[\frac{1}{2}\]
B) 1
C) \[2\]
D) None of these
Correct Answer: A
Solution :
\[\underset{x\to \infty }{\mathop{\lim }}\,\,\sqrt{{{a}^{2}}{{x}^{2}}+ax+1}-\sqrt{{{a}^{2}}{{x}^{2}}+1}\] \[=\underset{x\to \infty }{\mathop{\lim }}\,\,\frac{ax}{\,\sqrt{{{a}^{2}}{{x}^{2}}+ax+1}+\sqrt{{{a}^{2}}{{x}^{2}}+1}}\] \[=\underset{x\to \infty }{\mathop{\lim }}\,\,\frac{a}{\,\sqrt{{{a}^{2}}+\frac{a}{x}+\frac{1}{{{x}^{2}}}}+\sqrt{{{a}^{2}}+\frac{1}{{{x}^{2}}}}}=\frac{a}{2a}=\frac{1}{2}\].
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