Question

whats the limit of the following function?

Answer 1

\[\frac{ x ^{2}-\sqrt{x} }{ 1-\sqrt{x} }\]

Answer 2

x-->1

Answer 3

Do you know how to solve this?

Answer 4

you may try rationalizing, both numerator and denom separately..

Answer 5

but how can i do that? do first the denomitador and then de numerator or both at the same time?

Answer 6

doesnt matter..

Answer 7

any order you prefer..

Answer 8

??

Answer 9

The limit is -3.

Answer 10

you may rationalize either denom or nume first,,order wont matter..

Answer 11

Use L'Hospital Rule

Answer 12

mm but how? maybe something like this? \[\frac{ x ^{2} +\sqrt{x} }{ x ^{2} +\sqrt{x} }\] multiply the original function times the rationalization of the numerator?

Answer 13

i get 3x

Answer 14

yeep, i know the limit, but what i dont, is how to find it

Answer 15

Just take the derivative of the top, and then take the derivative of the bottom,

Answer 16

It is -3

Answer 17

Then plug in 1 for x and that will give you the limit.

Answer 18

mm but how can i find the limit without using L'hopital? cos my teacher doesnt allow me to do that yet, he want me to find the limit algebraically

Answer 19

:/

Answer 20

Just multiply top and bottom by conjugates then.

Answer 21

respectively? i mean the numerator conjugate times the original numerator and the denominator conjugate times the original denominator?

Answer 22

no,

Answer 23

What you want to do is rationalize one side.

Answer 24

So, just multiply top and bottom by 1+sqrt(x)

Answer 25

You should get 1-x for the bottom

Answer 26

\[\frac{ x ^{2}-\sqrt{x}(1+\sqrt{x}) }{1-x }\] so thts my result after doing what u said :p is it correct?

Answer 27

you would have parenthesis around x^2 and sqrt x

Answer 28

mm yes so what can i do next?

Answer 29

I am not sure actually you still have on the denominator 1-1 which is still dividing by a zero

Answer 30

your limit does not exist?

Answer 31

\(\huge\frac{ (x ^{2}-\sqrt{x})(1+\sqrt{x}) }{1-x }\times \frac{x^2+\sqrt x}{x^2+\sqrt x}\)

Answer 32

x^4-x = x(x^3-1) = x(x-1)(x^2x+x+1)

Answer 33

put x=1 after cancelling 1-x

Answer 34

divide both num and dem by x^2..