What is #lim x->0# of #(sqrt(x+4)-sqrt4)/x#?

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Answer 1 · 2016-08-15T02:58:54

#lim_(x ->0) (sqrt(x + 4) - sqrt(4))/x = 1/4#

Rationalize the numerator:

#=lim_(x ->0)(sqrt(x + 4) - sqrt(4))/x xx (sqrt(x + 4) + sqrt(4))/(sqrt(x + 4) + sqrt(4))#

#=lim_(x ->0) (x + 4 + 2sqrt(x + 4) - 2sqrt(x + 4) - 4)/(xsqrt(x + 4) + 2x)#

#=lim_(x ->0)(x)/(xsqrt(x + 4) + 2x)#

Factor out an x:

#=lim_(x -> 0)x/(x(sqrt(x + 4) + 2))#

#=lim_(x ->0) 1/(sqrt(x + 4) + 2)#

#=1/(sqrt(4) + 2)#

#= 1/4#

Hopefully this helps!