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lim-x-sin-x-1-sin-x-




Question Number 158204 by tounghoungko last updated on 01/Nov/21
  lim_(x→∞) (sin (√(x+1))−sin (√(x ))) =?
limx(sinx+1sinx)=?
Answered by ajfour last updated on 01/Nov/21
L=lim_(x→∞) 2cos ((((√x)+(√(x+1)))/2))sin ((((√(x+1))−(√x))/2))  =lim_(h→0) 2cos ((((1/( (√h)))+((√(h+1))/( (√h))))/2))sin (((((√(h+1))−1)/( (√h)))/2))  =lim_(h→0) 2cos (((2+h/2)/( 2(√h))))sin (((√h)/4))  L=(1/2)lim_(h→0) (√h)cos ((1/( (√h))))  =((1/2))((((1/(2(√h)))))/(−(1/(2h(√h))) sec ((1/( (√h))))tan ((1/( (√h))))))  L=−(L^2 /2)(1/(sin ((1/( (√h))))))=−(L^2 /(2(√(1−((4L^2 )/h)))))  ⇒   L=0  .
L=lim2cosx(x+x+12)sin(x+1x2)=lim2cosh0(1h+h+1h2)sin(h+11h2)=lim2cosh0(2+h/22h)sin(h4)L=12limh0hcos(1h)=(12)(12h)12hhsec(1h)tan(1h)L=L221sin(1h)=L2214L2hL=0.
Answered by cortano last updated on 01/Nov/21

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