How do you evaluate the limit #(sqrt(x+25)-5)/x# as x approaches #0#?

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Answer 1 · 2016-08-21T13:47:53

#= 1/10 #

#lim_(x to 0) (sqrt(x+25)-5)/x#

#= lim_(x to 0) (5sqrt(1+x/25)-5)/x#

and by Binomial Expansion of the following expression

#= lim_(x to 0) (5(1+x/25)^(1/2)-5)/x#

we have
#= lim_(x to 0) (5(1+(1/2) (x/25) + O (x^2) ) - 5) /x #

#= lim_(x to 0) ( x/10 + O (x^2)) /x #

#= lim_(x to 0) 1/10 + O (x) #

#= 1/10 #