limit question find answer

Question

$\lim_{x\rightarrow\infty}(\sqrt{x}-x)=?$

Comments

It is 0.

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it will tend to $-\infty$

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Initially it was leading to 0 :p

If tending to infinity then it must be -infinity

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How  to solve?

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$\lim_{x->\infty }(\sqrt{x}-x)$

let $x=\frac{1}{y}$ so that as  $x->\infty$  $y->0^{+}$

$\lim_{y->0^{+}}\frac{1}{\sqrt{y}}-\frac{1}{y}$

$\lim_{y->0^{+}}\frac{y-\sqrt{y}}{y\sqrt{y}}$

apply L-hospital since $\frac{0}{0}$ form.

$\lim_{y->0^{+}}\frac{1-\frac{1}{2\sqrt{y}}}{\sqrt{y}+\frac{\sqrt{y}}{2}}$

apply $y->0^{+}$

$\frac{1-\infty }{0+0}$   $-> -\infty$

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joshi_nitish beautifully explained but can we trust the intuition from the question itself that it will be -∞ 

because root(∞) - ∞ will be -∞

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YES if you search on the net you will find solutions for such problems solved by intuition. But in Gate maximum times intuitions might go wrong so better to try with exact method .

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better approach is to go with effective degree

$\sqrt{x} - x$   effective degree  $\sqrt{x} = 1/2$ 

effective degree of x = 1 

effective degree not getting cancelled we can use method now see  in whole expression  of $(x^{\frac{1}{2}} - x )$ 

x will dominate  then on putting  x-> -∞  in x we get  -∞  as answer , but since  - ∞ not a number i say answer = does not exist

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ok thanks :)

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@Mk Utkarsh  bro wrong approach   root(∞) - ∞ will be -∞    ( no its not )

root ( ∞) = still ∞  

∞ - ∞   (Indeterminant form )   you cannot say its = -∞

∞ + ∞ = ∞ ( its valid ) 

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life is tough :p thanks to you too

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  yup it is  @Mk Utkarsh 

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I think anwer is -1.

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how ?