Menu Close

The-largest-value-of-non-negative-integer-a-for-which-lim-x-1-ax-sin-x-1-a-x-sin-x-1-1-1-x-1-x-1-4-is-




Question Number 160065 by Kunal12588 last updated on 24/Nov/21
The largest value of non-negative integer a  for which lim_(x→1) {((−ax+sin(x−1)+a)/(x+sin(x−1)−1))}^((1−x)/( 1−(√x))) =(1/4)  is ........?
Thelargestvalueofnonnegativeintegeraforwhichlimx1{ax+sin(x1)+ax+sin(x1)1}1x1x=14is..?
Commented by Kunal12588 last updated on 24/Nov/21
it gives a=0,2 but they are saying a=2 is invalid  How is a=2 invalid?
itgivesa=0,2buttheyaresayinga=2isinvalidHowisa=2invalid?
Commented by Kunal12588 last updated on 24/Nov/21
he is a tutor. I think he is wrong that′s why   I asked.  his reasoning was L=lim_(x→1) (((1−a)/2))^(1+(√x))   and for a=2, L=lim_(x→1) (−(1/2))^(1+(√x))   but (negative)^((odd)/(even)) =not definedfor ∈R  so as x is approaching 1, L will be (−(1/2))^((1999...)/(100...))   which should not defined for real numbers
heisatutor.IthinkheiswrongthatswhyIasked.hisreasoningwasL=limx1(1a2)1+xandfora=2,L=limx1(12)1+xbut(negative)oddeven=notdefinedforRsoasxisapproaching1,Lwillbe(12)1999100whichshouldnotdefinedforrealnumbers
Commented by Kunal12588 last updated on 24/Nov/21
so what do you think? a=2 is valid or not?
Commented by mr W last updated on 24/Nov/21
i changed my mind. they are right.  see below.
ichangedmymind.theyareright.seebelow.
Commented by Kunal12588 last updated on 24/Nov/21
sir does this mean when a = 2, L≠(1/4)
sirdoesthismeanwhena=2,L14
Commented by Kunal12588 last updated on 24/Nov/21
Commented by mr W last updated on 24/Nov/21
for x, y ∈R, with a=2  y=f(x)=[((sin (x−1)−2(x−1))/(sin (x−1)+(x−1)))]^(1+(√x))   is not defined. so limit lim_(x→1) f(x)  doesn′t exist.    in wolfram alpha, y∈C.    therefore you should define if you  have to do with real functions or  with complex functions.
forx,yR,witha=2y=f(x)=[sin(x1)2(x1)sin(x1)+(x1)]1+xisnotdefined.solimitlimx1f(x)doesntexist.inwolframalpha,yC.thereforeyoushoulddefineifyouhavetodowithrealfunctionsorwithcomplexfunctions.
Commented by Kunal12588 last updated on 24/Nov/21
thank you sir
thankyousir
Answered by mr W last updated on 24/Nov/21
t=x−1  L=lim_(t→0) {((sin t−at)/(sin t+t))}^(1+(√(t+1)))   L=(((1−a)/2))^2   ((1−a)/2)≥0 ⇒a≤1           ∗)  (((1−a)/2))^2 =(1/4)  ⇒a=0, 2 (rejected due to ∗))    ∗)  f(t)^(g(t))  is defined only for f(t)≥0  if g(t)≠even
t=x1L=limt0{sintatsint+t}1+t+1L=(1a2)21a20a1)(1a2)2=14a=0,2(rejecteddueto)))f(t)g(t)isdefinedonlyforf(t)0ifg(t)even

Leave a Reply

Your email address will not be published. Required fields are marked *