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  • , where n is a positive integer or any positive rational, is  </p

\mathrm{f(x)=x^n}, where n is a positive integer or any positive rational, is

 

Option: 1

continuous
 


Option: 2

 discontinuous
 


Option: 3

 cannot be determined
 


Option: 4

 None of these


Answers (1)

best_answer

 Let x=c be any value of x.
Then, \mathrm{f(c+0)=\lim _{h \rightarrow 0}(c+h)^n=c^n }
and  \mathrm{f(c-0)=\lim _{h \rightarrow 0}(c-h)^n=c^n }
Also, \mathrm{f(c)=c^n }
\mathrm{\therefore \quad f(c+0)=f(c-0)=f(c) }
Hence, \mathrm{f(x)=x^n\, is\, \, continuous\, \, for\, x=c }
But \mathrm{c} is any value of \mathrm{x}. Therefore, \mathrm{f(x)=x^n } is continuous for all values of \mathrm{x}.

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vinayak

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