What is the limit of #(sqrt(x+5) - 3)/(x - 4)# as x approaches 4?

2 Answers
May 18, 2015

Let's multiply both numerator and denominator of this expression by #sqrt(x+5)+3# to get rid of undefined #0/0# value.

Assuming #x!=4# in our expression,

#(sqrt(x+5)-3) / (x-4) = #

#= [(sqrt(x+5)-3) * (sqrt(x+5)+3)] / [(x-4)*(sqrt(x+5)+3)] =#

#= (x+5-9) / [(x-4)*(sqrt(x+5)+3)] =#

#= (x-4)/ [(x-4)*(sqrt(x+5)+3)] =#

#= 1/ (sqrt(x+5)+3)#

As #x->4#, this expression tends to #1/(sqrt(4+5)+3)=1/6#.

Therefore,

#lim_(x->4)(sqrt(x+5)-3) / (x-4) = 1/6#

Nov 1, 2015

#frac{1}{6}#

Explanation:

#lim_{x->4}(sqrt{x+5}-3)=0#

#lim_{x->4}(x-4)=0#

Using the L'Hospital Rule,

#lim_{x->4}frac{sqrt{x+5}-3}{x-4}#

#=lim_{x->4}frac{frac{d}{dx}(sqrt{x+5}-3)}{frac{d}{dx}(x-4)}#

#=lim_{x->4}frac{(frac{1}{2sqrt{x+5}})}{1}#

#=frac{1}{2sqrt{4+5}}#

#=1/6#