Expert: Ahmed Salami - 8/30/2010

Question
Hi Ahmed...so I have tried this question a million different ways and I emailed my math teacher who helped me enough so that I know the answer is 1/6. But I can't understand her instructions and she emailed me from Greece (I feel bad asking again for her to clarify). Can you? The question says: evaluate the limit as x approaches 3+ of [√(x-3)]/[x^2-9]. Please help! Thanks, Samara

Answer
Hi Samara,
The function has a singularity at x = 3 since x² - 9 = 0 and the expression is then undefined. It is simply clear from this that the line x = 3 is an asymptote to the curve and so the limit, if we approach from 3+, would be infinity.
Exploring it algebraically,
[√(x-3)]/(x^2 - 9) = [√(x-3)]/(x-3)(x+3)
= 1/(x+3)√(x-3)
The term √(x-3) approaches 0 from x = 3+ and so
lim(x->3+) [√(x-3)]/(x^2 - 9) = 1/0 = ∞

So, i'm afraid the answer is not even 1/6. You may check the written expressions again in case you missed something.

Regards