Discuss the continuity of f(x) where f(x) = lim n→∞ ( sin∏x∕2 )2n

Asked by yeeshuraj3 | 2nd Aug, 2016, 12:08: PM

Expert Answer:

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space space space space space space space space space space space space space space space space space space space space space space space space space space limit as straight n rightwards arrow infinity of x to the power of 2 n end exponent equals open curly brackets table attributes columnalign left end attributes row cell 0 comma space open vertical bar straight x close vertical bar less than 1 end cell row cell 1 comma space open vertical bar straight x close vertical bar equals 1 end cell end table close

S o comma space f left parenthesis x right parenthesis equals limit as straight n rightwards arrow infinity of open parentheses sin open parentheses πx over 2 close parentheses close parentheses to the power of 2 straight n end exponent equals open curly brackets table attributes columnalign left end attributes row cell 0 comma space open vertical bar s i n πx over 2 close vertical bar less than 1 end cell row cell 1 comma space open vertical bar s i n fraction numerator pi x over denominator 2 end fraction close vertical bar equals 1 end cell end table close

Thus space if space straight f left parenthesis straight x right parenthesis space is space continuous space for space all space straight x comma space except space space for space those space values space of space straight x space for space which space
open vertical bar sin πx over 2 close vertical bar equals 1 space comma space for space all space odd space integers space straight x equals open parentheses 2 straight n plus 1 close parentheses space where space straight n element of I

Checking space continuity space at space straight x equals open parentheses 2 straight n plus 1 close parentheses
LHL space equals space limit as straight x rightwards arrow 2 straight n plus 1 of space straight f left parenthesis straight x right parenthesis equals 0
and space straight f left parenthesis 2 straight n plus 1 right parenthesis space equals 1
Thus comma space LHL not equal to straight f left parenthesis 2 straight n plus 1 right parenthesis
rightwards double arrow straight f space is space not space continuous space at space straight x equals space 2 straight n plus 1
The space points space where space straight f left parenthesis straight x right parenthesis space is space discontinuous space is space straight x equals 2 straight n plus 1 space where space straight n element of I



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Answered by Vijaykumar Wani | 4th Aug, 2016, 11:01: AM