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