Limit w/o L'hopital

Question

$$\lim\limits_{x\to c} \dfrac{\sqrt{x}-\sqrt{c}}{\sqrt{x^2-c^2}} = ?$$

---

I tried factorizing and rationalizing numerator/denominator but no use. Any help/hint ? Thanks!

Answer 1

HINT:

$$\frac{\sqrt x-\sqrt c}{\sqrt{x^2-c^2}}=\frac{x-c}{(\sqrt x+\sqrt c)\sqrt{(x+c)(x-c)}}$$

Clearly, $x^2>c^2$

If $c>0, x>c>0$

If $c<0,x<c<0$

Answer 2

The simplest method is to define $t = x - c$. Then express your equation in terms of $t$. Now since $t$ goes to zero, we can simply expand the numerator and the denominator in a Taylor series. Taking the ratio of the lowest non-zero terms will yield the correct result. This method always works fine. There is no need to remember special tricks.