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  4. Let f (x) = (1 /√ 18-x2), ten value o f limx → 3 ( (f(x)-f(3) /x-3))

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Q. Let f (x) = $ \frac {1} {\sqrt {18-x^2}}$, ten value o f $ \lim_{x \to 3} (\frac {f(x)-f(3)} {x-3}) $

Solution:

$\lim _{x\to 3} \frac{\left(18 -x^{2}\right)^{-\frac{1}{2} }- \left(\frac{1}{3}\right)}{x-3} =\lim _{x\to 3} \frac{3 -\sqrt{18 -x^{2}}}{3\left(x-3\right) \sqrt{18 -x^{2}} } $
= $\lim _{x\to 3} \left[\frac{9-\left(18 -x^{2}\right)}{3\left(x-3\right) \sqrt{18 -x^{2}}\left(3+\sqrt{18 -x^{2}}\right)}\right]$
= $\lim _{x\to 3} \frac{x^{ 2} -9}{3\left(x-3\right) \sqrt{18 -x^{2} } \left[3+\sqrt{18 -x^{2}}\right]}$
= $\lim _{x\to 3} \frac{x+3}{3\sqrt{18-x^{2}} \left[3+\sqrt{18 -x^{2}}\right] }$
= $\frac{6}{3\left(3\right)\left(3+3\right)} =\frac{1}{9}$