InterviewSolution
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InterviewSolution
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`Lim_(x->oo) ((x/(x+1))^a + sin (1/x))^x` is equal toA. `e^a-1`B. `e^1-a`C. `e`D. `0` |
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Answer» Correct Answer - B We have `lim_(xto oo) {((x)/(x+1))^a a+sin.(1)/(x)}^x` `lim_(xto oo) {(1)/(x+(1)/(x))^a +sin.(1)/(x)}^x` ` lim_(xto oo) {1+(1+(1)/(x))^-a+ sin .(1)/(x)-1}^x` `= lim_(y to oo) {1+(1+y)^-a+siny 1}^(1//y)`, where `y-(1)/(x)` `=e^(lim_(xto0)((1+y)^-1+siny-1)/(y)` `=e^(lim_(yto0) {(siny)/(y)+((1+y)^-a)/(y)}` ` =e^(1-a)[because lim_(yto0) (siny)/(y)=1and lim_(yto0) ((1+y)^-a-1)/(y)=-a]` |
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