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

2 Answers
Mar 8, 2018

Answer:

The limit is #-1/3#

Explanation:

It is similar to both

#lim_(xrarr4)(3-sqrt(5+x))/(x-4)# and to #lim_(xrarr4)(x-4)/(1-sqrt(5-x))#

but both in one expression.

So multiply #(3-sqrt(5+x))/(1-sqrt(5-x))# by

#((3+sqrt(5+x)))/((3+sqrt(5+x))) * ((1+sqrt(5-x)))/((1+sqrt(5-x)))# to get:

#lim_(xrarr4)((9-(5+x))(1+sqrt(5-x)))/((3+sqrt(5+x))(1-(5-x))#

# = lim_(xrarr4)((4-x)(1+sqrt(5-x)))/((3+sqrt(5+x))(-(4-x))#

# = lim_(xrarr4)(-(1+sqrt(5-x)))/(3+sqrt(5+x))#

# = (-(1+sqrt1))/(3+sqrt9) = -2/6 = -1/3#

Mar 8, 2018

Answer:

The limit should approach -1/3, I screwed up the original answer.

Explanation:

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

first multiply the top and bottom by the conjugate of the numerator and the conjugate of the denominator

#(3-sqrt(5+x))/(1-sqrt(5-x))*#

#(3+sqrt(5+x))/(3+sqrt(5+x))*(1+sqrt(5-x))/(1+sqrt(5-x))#

# = (4-x)/(-(4-x))*(1+sqrt(5-x))/(3+sqrt(5+x)) #

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

plug in the limit value to get your answer:

# =-(1+sqrt(5-4))/(3+sqrt(5+4)) #
# =-(1+1)/(3+3) = -2/6 = -1/3 #

Tony B