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Lt          x^a/x+1 , where 0≤a<1
x  

Asked by rishabhnand10 20th December 2016, 8:56 PM
Answered by Expert
Answer:
begin mathsize 16px style Consider space the space given space expression Lt with straight x rightwards arrow infinity below fraction numerator straight x to the power of straight a over denominator straight x plus 1 end fraction
Clearly comma space the space degree space of space the space numerator space is space less space than space the space degree space of space the space denomintor. space space space left parenthesis Since space 0 less or equal than straight a less than 1 right parenthesis
So comma space in space this space case comma space we space will space divide space the space numerator space and space denominator space by space 1 over straight x. space left parenthesis Since space in space this space case space the space degree space of space the space denominator space is space greater space than space the space degree space of space the space numerator space right parenthesis
So space we space get comma
Lt with straight x rightwards arrow infinity below fraction numerator straight x to the power of straight a over denominator straight x plus 1 end fraction equals Lt with straight x rightwards arrow infinity below fraction numerator open parentheses straight x to the power of straight a close parentheses 1 over straight x over denominator open parentheses straight x plus 1 close parentheses 1 over straight x end fraction equals Lt with straight x rightwards arrow infinity below fraction numerator straight x to the power of straight a minus 1 end exponent over denominator 1 plus 1 over straight x end fraction
Now comma space we space can space see space that space
Now space in space this space case comma space the space bottom space is space going space to space 1 plus 0 equals 1 space and space the space top space is space going space to space infinity space left parenthesis because space of space the space lingering space straight x to the power of straight a minus 1 end exponent space term right parenthesis comma space
so space in space this space case space we space use space “ extended space limit space laws ” space. space
According space to space the space extended space limits space law comma
Lt with straight x rightwards arrow infinity below straight x to the power of straight k equals open curly brackets table row cell infinity comma space space space space if space straight k greater than 0 end cell row cell 1 comma space space space space space space if space straight k equals 0 end cell row cell 0 comma space space space space space space if space straight k less than 0 end cell end table close
Since space in space the space given space question comma space 0 less or equal than straight a less than 1 comma space that space is comma space left parenthesis straight a minus 1 right parenthesis space less than 1
Lt with straight x rightwards arrow infinity below fraction numerator straight x to the power of straight a minus 1 end exponent over denominator 1 plus 1 over straight x end fraction equals fraction numerator Lt with straight x rightwards arrow infinity below open parentheses straight x to the power of straight a minus 1 end exponent close parentheses over denominator Lt with straight x rightwards arrow infinity below open parentheses 1 plus 1 over straight x close parentheses end fraction equals fraction numerator 0 over denominator 1 plus 0 end fraction equals 0
end style
Answered by Expert 21st December 2016, 4:55 PM
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