lim sqrt (5-x)-1 / 2- sqrt (x)

Question

i need help, can you help me solve this answer. 

Answer 1

Hello Kevin & Phillips,

 

Once we apply L hopital's rule

We get the following----

 

[(-1/2)(5-x)^(-1/2)]/(-1/2)x^(-1/2)

 

on simplifying

 

x^(1/2)/(5-x)^(-1/2)

 

On taking limits x tending to 4

 

2/1=2

 

So the answer obtained by Kevin is correct in my opinion.

 

Answer 2

Another way, using algebra, is to multiply numerator and denominator by the conjugate of the numerator, thus rationalizing the numerator.

 

So the numerator multiplied by its conjugate:  (√(5-x) - 1)(√(5-x) + 1)  =  (5 - x) - 1 = 5 - x - 1 = 4 - x.

 

The fraction now becomes:  (4 - x)/((2 - √x)(√(5-x) + 1)).

 

Factor 4 - x as (2 - √x)(2 + √x) giving us:  ((2 - √x)(2 + √x))/((2 - √x)(√(5-x) + 1)).

 

Simplify and we get: (2 + √x)/(√(5-x) + 1).

 

Replace x by 4, and we get (2+2)/(√1 + 1) or 4/2 or 2.

Answer 3

if the limit is x approaches 4 then you cannot make a direct substitution into the expression because you would get 0/0.

 

I don't know whether you learned LHopitals rule yet but using it we take the first derivative of the numerator and the denominator and then evaluate the limit.

 

-1/2√(5-x) divided by -1/2√x

 

Now plug in 4 for x

 

-1/2 divided by -1/4

 

Therefore the limit is 2