appleduardo
  • appleduardo
whats the limit of the following function?
Mathematics
jamiebookeater
  • jamiebookeater
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appleduardo
  • appleduardo
\[\frac{ x ^{2}-\sqrt{x} }{ 1-\sqrt{x} }\]
appleduardo
  • appleduardo
x-->1
anonymous
  • anonymous
Do you know how to solve this?

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shubhamsrg
  • shubhamsrg
you may try rationalizing, both numerator and denom separately..
appleduardo
  • appleduardo
but how can i do that? do first the denomitador and then de numerator or both at the same time?
shubhamsrg
  • shubhamsrg
doesnt matter..
shubhamsrg
  • shubhamsrg
any order you prefer..
appleduardo
  • appleduardo
??
anonymous
  • anonymous
The limit is -3.
shubhamsrg
  • shubhamsrg
you may rationalize either denom or nume first,,order wont matter..
anonymous
  • anonymous
Use L'Hospital Rule
appleduardo
  • appleduardo
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?
anonymous
  • anonymous
i get 3x
appleduardo
  • appleduardo
yeep, i know the limit, but what i dont, is how to find it
anonymous
  • anonymous
Just take the derivative of the top, and then take the derivative of the bottom,
anonymous
  • anonymous
It is -3
anonymous
  • anonymous
Then plug in 1 for x and that will give you the limit.
appleduardo
  • appleduardo
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
appleduardo
  • appleduardo
:/
anonymous
  • anonymous
Just multiply top and bottom by conjugates then.
appleduardo
  • appleduardo
respectively? i mean the numerator conjugate times the original numerator and the denominator conjugate times the original denominator?
anonymous
  • anonymous
no,
anonymous
  • anonymous
What you want to do is rationalize one side.
anonymous
  • anonymous
So, just multiply top and bottom by 1+sqrt(x)
anonymous
  • anonymous
You should get 1-x for the bottom
appleduardo
  • appleduardo
\[\frac{ x ^{2}-\sqrt{x}(1+\sqrt{x}) }{1-x }\] so thts my result after doing what u said :p is it correct?
anonymous
  • anonymous
you would have parenthesis around x^2 and sqrt x
appleduardo
  • appleduardo
mm yes so what can i do next?
anonymous
  • anonymous
I am not sure actually you still have on the denominator 1-1 which is still dividing by a zero
anonymous
  • anonymous
your limit does not exist?
hartnn
  • hartnn
\(\huge\frac{ (x ^{2}-\sqrt{x})(1+\sqrt{x}) }{1-x }\times \frac{x^2+\sqrt x}{x^2+\sqrt x}\)
hartnn
  • hartnn
x^4-x = x(x^3-1) = x(x-1)(x^2x+x+1)
hartnn
  • hartnn
put x=1 after cancelling 1-x
linknissan
  • linknissan
divide both num and dem by x^2..

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