Calculus Limit Question -my ignorance is astounding?

Question

ok could someone guide me through solving the Limit, finally posted the correct problem
Evaluate the Limit.

Lim........(SqRoot( 6 - x ) - 2)
x->2...__ __ __ __ __ __ __ __
.............(SqRoot( 3 - x ) - 1)


so lim (SqRoot(6-x) -2) / (SqRoot(3-x) - 1)
...x-> 2

erm..we havnt exactly learned about derivitives yet, but the answer is .5 when i check with a calculator

is this possible to solve w/o derivitives?

Answer 1

No, plug 2.1 in for x, and see what you come up with. (0.5)

Answer 2

The limit evaluates to: (2 - 2)/(1 - 1) = 0/0

This, in calculus terms is inconclusive, so you have to use L'Hopital's Rule, which is to take the derivative of the numerator and denominator separately and then reapply the limit.

Numerator derivative: -(1/2) (6 - x)^(-1/2)

Demoninator derivative: -(1/2) (3 - x)^(-1/2)

Now apply the rule:

==> [-(1/2) (6 - x)^(-1/2)]/[ -(1/2) (3 - x)^(-1/2)]
==> [(3 - x)^(1/2)]/[(6 - x)^(1/2)]
==>(1/4)/(1/2)
==> 2/4
==> 1/2

So, the limit evaluates to "1/2".

Make sense? Was it clear enough? Let me know if you don't understand this, but this technique os usaully very well documented in textbooks.


:)

Answer 3

This expression is 0/0 at x=2, so use L'hospital's rule, take the derivative of the numerator and denominator. (You ignore the constant terms in each.)

d[(6-x)^(1/2)]/dx = .5*(6-x)^(-1/2)*(-1) = -.5*(6-x)^(-1/2)

d[(3-x)^(1/2)]/dx = .5*(3-x)^(-1/2)*(-1) = -.5*(3-x)^(-1/2)

The resulting expression is

(6-x)^(-1/2) / (3-x)^(-1/2)

Evaluate at x=2 to get 4^(-1/2)/1^(-1/2) or 4^-(1/2) = .5

Answer 4

Input 2 for x and see what you come up with. Remember a zero in the denominator tends to infinity.

Answer 5

Use L'Hospital's rule. Differentiate the numerator and the denominator with respect to x and then you can plug in x = 2 and whatever you get is the limit.

lim (x -> 2) [(6 - x)^0.5 -2]/[(3 - x)^0.5 - 1]
= lim (x -> 2) [-0.5*(6 - x)^(-0.5)]/[-0.5*(3 - x)^(-0.5)]
= 0.25/0.5
= 0.5

Good luck!

=)

Answer 6

I'm not sure of my math but I get 1/2.
Take the derivative of the numerator and denomenator. This is based on the property that.
Lim x/y = Lim dx/dy

Lim.........(6-x)^1/2 - 2
x->2.......--------------------
..............(3-x)^1/2 - 1

Lim.........1/2*(6-x)^-1/2*-1
x->2........---------------------
...............1/2*(3-x)^-1/2*-1

Simplifying by multiply by (-2/-2)

Lim........(6-x)^-1/2
x->2......--------------
.............(3-x)^-1/2

Getting rid of the ^-1/2

Lim.......(3-x)^1/2
x->2......-------------
.............(6-x)^1/2

Inserting x=2

(3-2)^1/2 1
---------- = ---------
(6-x)^1/2 2

Answer 7

gp4arts is the first correct answer. Use leHopital's rule.