How to show work for this limit (calculus)

The limit as x→3 of ( √(25-x^2) - 4) / (x-3)

I know from looking at the graph on my calculator that the answer is -3/4 but how would I actually explain that if something like this was on a quiz (we have one on this stuff Tuesday without calculator use so that's why I'm asking you guys)?

Thanks in advance!

Use the conjugate.
lim(x→3) [√(25 - x^2) - 4] / (x - 3)
= lim(x→3) [√(25 - x^2) - 4][√(25 - x^2) + 4] / {(x - 3) [√(25 - x^2) + 4]}
= lim(x→3) [(25 - x^2) - 16] / {(x - 3) [√(25 - x^2) + 4]}
= lim(x→3) (9 - x^2) / {(x - 3) [√(25 - x^2) + 4]}
= lim(x→3) -(x - 3)(x + 3) / {(x - 3) [√(25 - x^2) + 4]}
= lim(x→3) -(x + 3) / [√(25 - x^2) + 4]
= -(3 + 3)/(4 + 4)
= -3/4.

I hope this helps!

Your parentheses are unbalanced so I am guessing that you meant
√((25-x^2) - 4) / (x-3).
Either way there is vertical asymptote at x=3 so the limit does not exist.

See http://www.wolframalpha.com/input/?i=plo…
for the graph.

What you may be seeing is an anomoally on a graphing calculator is you set the window size inappropriately.