What is the result of lim x → ∞ (-x-1)/((x-1)(x-Sqrt(x^2+x+1)) ?

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Answer 1 · 2018-07-21T11:45:00

#2#

#lim x → ∞ (-x-1)/((x-1)(x-sqrt(x^2+x+1))#

#= lim x → ∞ (-x-1)/((x-1)(x-sqrt((x+ 1/2)^2+3/4))#

#y = x + 1/2 qquad x = y - 1/2#

#= lim y → ∞ (-y-1/2)/((y-3/2)(y - 1/2-sqrt(y^2+3/4))#

#= lim y → ∞ (-1-1/(2y))/((y-3/2)(1 - 1/(2y)-sqrt(1+3/(4y^2)))#

By binomial expansion:

#sqrt(1+3/(4y^2) )= 1 + (1/2)3/(4y^2) + bbbO(1/y^6) #

#= lim y → ∞ (-1-1/(2y))/((y-3/2)(1 - 1/(2y)-1 - (3/(8y^2)) ))#

#= lim y → ∞ (1+1/(2y))/((y-3/2)( 1/(2y) + 3/(8y^2) ))#

#= lim y → ∞ (1+cancel(1/(2y)))/(1/2 + cancel(3/(8y)) - cancel(3/(4y)) - cancel(9/(16y^2)))#

#= 2#

Answer 2 · 2018-07-21T12:47:30

#2#

#\lim_{x\to \infty} \frac{-x-1}{(x-1)(x-\sqrt{x^2+x+1})}#

#=\lim_{x\to \infty} \frac{-(x+1)(x+\sqrt{x^2+x+1})}{(x-1)(x-\sqrt{x^2+x+1})(x+\sqrt{x^2+x+1})}#

#=\lim_{x\to \infty} \frac{-(x+1)(x+\sqrt{x^2+x+1})}{(x-1)(x^2-(\sqrt{x^2+x+1})^2)}#

#=\lim_{x\to \infty} \frac{-(x+1)(x+\sqrt{x^2+x+1})}{(x-1)(x^2-x^2-x-1)}#

#=\lim_{x\to \infty} \frac{-(x+1)(x+\sqrt{x^2+x+1})}{-(x-1)(x+1))#

#=\lim_{x\to \infty} \frac{x+\sqrt{x^2+x+1}}{x-1)#

#=\lim_{x\to \infty} \frac{x+x\sqrt{1+1/x+1/x^2}}{x(1-1/x))#

#=\lim_{x\to \infty} \frac{1+\sqrt{1+1/x+1/x^2}}{1-1/x)#
# =\frac{1+\sqrt{1+0+0}}{1-0)#

#=\frac{2}{1}#

#=2#