As the limit x approaches 9, 9-x/3-sqrt(x)...can someone help me figure this out?

Question

I have tried the conjugate method and other things and still can't seem to find the answer...does anyone have any suggestions or answers!? THANKS!

Answer 1

This is a limit of the kind lim x -> a (a - x)/(sqrt(a) - sqrt(x)). Multiply the numerator and the denominator by sqrt(a) + sqrt(x). You get

lim (x -->a) [(a-x ) (sqrt(a) + sqrt(x)]/((a - x)] = lim (x --> a) sqrt(a) + sqrt(x). Now, we have no indetermination and our limit is sqrt(a) + sqrt(a) = 2 sqrt(a). In your case, a = 9, so the limit is 6.

If you have studied derivatives, you know this just the inverse of the derivative of sqrt(x)

Answer 2

Which course are you taking? An easy way to do it would be using L'Hopital's rule if you're in Calculus.

Then since the top and the bottom both approach 0, take the derivative of the top and of the bottom. Then you get

lim as x approaches 9 of (-1)/[-(1/2)(1/sqrt(x))]

Now you can just plug in 9, so it's

(-1)/[-(1/2)(1/sqrt(9))] = -1/(-1/6) = 6

Answer 3

You can also do it this way, which I think is easier. Multiply both top and bottom of equations by (3+sqrt(x))

that gives you
((9-x)(3+sqrt(x)))/((3-sqrt(x))(3+sqrt(x)))

Simplifying gives you
((9-x)(3+sqrt(x)))/(9-x)

Simplifying gives you
3+sqrt(x)

Put 9 in for x and you get
3+3 = 6

Answer 4

(9-x)/(3-√x)

write 9 - x = 3^2 - (√x)^2 = (3 + √x)(3 - √x)

substituting in the given expression

(3 + √x)(3 - √x)/(3 - √x)

=>(3 + √x)

when x ->9 , (3 +√9)

=>3 + 3 = 6

Answer 5

l'Hôpital's rule works well on this problem.

Another way you can solve this is to use the fact that 9 - x can be factored into (3 - sqrt(x)) * (3 + sqrt(x)) and then cancel out (3 - sqrt(x))

Good luck.

Answer 6

lim (9-x) / (3-sqrt(x))
x--> 9

Use L'Hopitals Rule since you have an indeterminant form of [zero / zero] when you take the limit.

That means take the derivative of the numerator and denominator individually.

lim (-1) / (-.5x^(-.5))
x--> 9

= 1 / [(.5)(9^(-.5))]
= 2 / (9^(-.5))
= 2 * 9^.5
= 2 * 3
= 6

Answer 7

lim (9-x)/(3-sqrt(x)) = 1
x--> 9