A) \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] exists for \[n>1\]
B) \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] exists for \[n<0\]
C) \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] Does not exist for any value of n
D) \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] cannot be determined
Correct Answer: A
Solution :
[a] for \[n>1,\] \[\underset{x\to 0}{\mathop{\lim }}\,{{x}^{n}}\sin (1/{{x}^{2}})=0x\] (any value between -1 and 1)=0 For n<0, \[\underset{x\to 0}{\mathop{\lim }}\,{{x}^{n}}\sin (1/{{x}^{2}})=\infty \times \] (any value between -1 and 1) = \[\infty \].
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