i've got a very simple exercise of limit, and i started solving it, but it won't get out of indetermination.

The limit is $$\lim_{x\to 1} = \frac{x^2-\sqrt x}{1-\sqrt x}$$

If i use L'Hôpital's it works, but just using basic math rules, it falls in another indetermination ($\frac{0}{0}$).

  • $\begingroup$ What are "basic math rules"? $\endgroup$ – Irregular User May 25 '16 at 17:55
  • $\begingroup$ root extraction is one of them. $\endgroup$ – Edilson May 25 '16 at 17:56

$$\frac{x^2-\sqrt x}{1-\sqrt x}=\frac{(x^2-\sqrt x)(1+\sqrt x)}{1-x}\;\color{red}{(**)}$$

But also

$$x^2-\sqrt x=\frac{x^4-x}{x^2+\sqrt x}=x\frac{\overbrace{(x-1)(x^2+x+1)}^{=x^3-1}}{x^2+\sqrt x}$$


$$\color{red}{(**)}=-\frac{x(x^2+x+1)(1+\sqrt x)}{x^2+x}\xrightarrow[x\to1]{}-\frac{3\cdot2}{2}=-3$$

Directly with l'Hospital:

$$\lim_{x\to1}\frac{x^2-\sqrt x}{1-\sqrt x}=\lim_{x\to1}\frac{2x-\frac1{2\sqrt x}}{-\frac1{2\sqrt x}}=\lim_{x\to1}\left(-4x^{3/2}+1\right)=-3$$

  • $\begingroup$ i was doing something wrong with the signal $\endgroup$ – Edilson May 25 '16 at 18:04
  • $\begingroup$ @Edilson Perhaps so, yet with l'Hospital it is simple, too. $\endgroup$ – DonAntonio May 25 '16 at 18:12
  • $\begingroup$ Yeah i know, but i wanted solve it using both methods, 'cause i won't get stuck in only one method. $\endgroup$ – Edilson May 25 '16 at 18:14
  • $\begingroup$ Once more, thank you. $\endgroup$ – Edilson May 25 '16 at 18:15
  • 1
    $\begingroup$ @Edilson Any time and nice going, yet once you go l'Hospital you don't really want to come back...unless you're forced to. $\endgroup$ – DonAntonio May 25 '16 at 18:15

Notice that:

$$\frac{x^2-\sqrt{x}}{1-\sqrt{x}}=\frac{\left(x^2-\sqrt{x}\right)\left(1+\sqrt{x}\right)}{\left(1-\sqrt{x}\right)\left(1+\sqrt{x}\right)}=\frac{\sqrt{x}\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)\left(x+\sqrt{x}+1\right)}{\left(1-\sqrt{x}\right)\left(1+\sqrt{x}\right)}=$$ $$-\sqrt{x}\left(x\sqrt{x}+1\right)=-\sqrt{x}-x-x^{\frac{3}{2}}$$

  • $\begingroup$ okay, thank you for the answer, but can you explain how did you get this ? $\endgroup$ – Edilson May 25 '16 at 17:59
  • $\begingroup$ @Edison: Set $t=\sqrt x$, and simplify the rational function in $t$ thus obtained. $\endgroup$ – Bernard May 25 '16 at 18:04
  • $\begingroup$ @Bernard Also a nice way, thanks for extra hint. $\endgroup$ – Jan May 25 '16 at 18:05
  • $\begingroup$ ... i'm getting trouble using this regex. $\endgroup$ – Edilson May 25 '16 at 18:10
  • $\begingroup$ Equality is true if $x \ne 1$. $\endgroup$ – student forever May 25 '16 at 18:29

if $t=\sqrt x$

$\lim_{x\to1}\frac{x^2-\sqrt x}{1-\sqrt x}=\lim_{t\to1}\frac{t^4-t}{1-t}=\lim_{t\to1}\frac{(t^3+t^2+t)(t-1)}{1-t}=-3$


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