How do you find the limit of #1/(root3(x^2-4)# as x approaches #2^-#?

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Answer 1 · 2016-07-09T09:36:47

# = - oo#

#lim_{x to 2^-} 1/(root3(x^2-4)#

let #x = 2 - h, 0 < h "<<" 1#

#implies lim_{h to 0} 1/(root3((2-h)^2-4)#

#= lim_{h to 0} 1/(root3(4 - 4h + h^2-4)#

#= lim_{h to 0} 1/(root3(- 4h + h^2)#

#= lim_{h to 0} (- 4h + h^2)^(-1/3)#

#= lim_{h to 0} h^(-1/3) (- 4 + h)^(-1/3)#

#= lim_{h to 0} - h^(-1/3) (4 - h)^(-1/3)#

#= lim_{h to 0} - (4h)^(-1/3) (1 - h/4)^(-1/3)#

by Taylor expansion

#= lim_{h to 0} - 1/root3(4h) (1 - (- 1/3) h/4 + O(h^2) )#

#= lim_{h to 0} - 1/root3(4h) (1 + h/12 + O(h^2) )#

# = - oo#