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

Question

Isaiah Allen · Accepted Answer

The limit is #-1/3#

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#

William Abdalla · Answer

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

#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 #

Christopher Allers · Answer

undefined The limit of (3 - sqrt(5 + x))/(1 - sqrt(5 - x)) as x approaches 4 is 2.

Emilia Aguilar · Answer

undefined To find the limit of the given function as x approaches 4:

lim(x→4) (3 - √(5 + x))/(1 - √(5 - x))

We first need to observe that as x approaches 4, both denominators of the fractions will approach 0. This suggests that we may need to rationalize the fractions to simplify the expression.

Multiplying the numerator and denominator of the fraction by the conjugate of the denominator to rationalize:

lim(x→4) [(3 - √(5 + x))(1 + √(5 - x))]/[(1 - √(5 - x))(1 + √(5 - x))]

Expanding the numerator and denominator:

lim(x→4) [3(1) + 3√(5 - x) - √(5 + x) - (5 - x)] / [1 - (5 - x)]

Simplify:

lim(x→4) [3 + 3√(5 - x) - √(5 + x) - 5 + x] / [6 - x]

Combine like terms:

lim(x→4) [x - 2 + 3√(5 - x) - √(5 + x)] / [6 - x]

Now, we can directly substitute x = 4 into the expression:

[4 - 2 + 3√(5 - 4) - √(5 + 4)] / [6 - 4]

[2 + 3√1 - √9] / 2

[2 + 3 - 3] / 2

2/2

1

So, lim(x→4) (3 - √(5 + x))/(1 - √(5 - x)) = 1.