How do you find the limit of #(sqrt(x+5)-3)/(x-4)# as #x->4#?

1 Answer
Steve M
Dec 9, 2016

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

Explanation:

# lim_(x rarr 4) (sqrt(x+5)-3)/(x-4) = lim_(x rarr 4) (sqrt(x+5)-3)/(x-4) * (sqrt(x+5)+3)/(sqrt(x+5)+3)#
# " " = lim_(x rarr 4) ((sqrt(x+5)-3))/((x-4)) * ((sqrt(x+5)+3))/((sqrt(x+5)+3))#
# " " = lim_(x rarr 4) (sqrt(x+5)^2-3^2)/((x-4)(sqrt(x+5)+3))#
# " " = lim_(x rarr 4) (x+5-9)/((x-4)(sqrt(x+5)+3))#
# " " = lim_(x rarr 4) (x-4)/((x-4)(sqrt(x+5)+3))#
# " " = lim_(x rarr 4) 1/(sqrt(x+5)+3)#
# " " = 1/(sqrt(4+5)+3)#
# " " = 1/(sqrt(9)+3)#
# " " = 1/(3+3)#
# " " = 1/6#

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