Question

Now what?

\[\Large \mathop {\lim }\limits_{x \to 3} {\rm{ \;\;log}}\frac{{x - 3}}{{\sqrt {x + 6} - 3}}\]

Answer 1

Do you know Calculus and L'Hôpital's Rule ?
See http://en.wikipedia.org/wiki/L'Hôpital's_rule

If you do, take the derivative of the numerator and denominator, and evaluate each at x=3
\( \frac{d}{dx}(x-3)= 1\)

\( \frac{d}{dx} (x+6)^{\frac{1}{2}} -3= \frac{1}{2\sqrt{x+6}}\)
and we have
\[ \frac{1}{\frac{1}{2\sqrt{x+6}}}= 2\sqrt{x+6} \]
As x \( \to \) 3 this approaches 6