I'm stumped on yet another assignment problem. I'm not allow to use power rule with this problem so i have to rely on good old
$$ \frac{f(a+h)-f(a)}{h} $$
so here are the steps ive taken thus far but i cant quite bring it home.
1-
$$\lim_{h\to 0}\frac{\frac{1}{\sqrt{t+h}}- \frac{1}{\sqrt{t}}}{h} $$
2- get common denominator $ \sqrt{t} \sqrt{t+h} $
$$\lim_{h\to 0}\frac{\frac{\sqrt{t}}{\sqrt{t+h}} – \frac{\sqrt{t+h}}{\sqrt{t}}}{h} $$
3- multiply by conjugate pair
$$\lim_{h\to 0}\frac{\frac{\sqrt{t}}{\sqrt{t+h}} – \frac{\sqrt{t+h}}{\sqrt{t}}}{h}* \frac{\sqrt{t}+\sqrt{t+h}}{\sqrt{t}+\sqrt{t+h}} $$
4-multiply across and cancel the h's and i end up with
$$ \frac{-1}{\sqrt{t+h}\sqrt{t}(\sqrt{t}+\sqrt{t+h} )}$$
this is where im stuck the solutions manual gets to
$\frac{-1}{\sqrt{t}\sqrt{t}(\sqrt{t}+\sqrt{t})} $
i have no idea how they could have achieved it? I'm missing an intermediate step can someone please point me in the right direction and i think my algebra is failing me here.
Best Answer
Since the square root function is continuous, then you can let the limit "pass through" the radicals, and take $h$ to $0$. We still have to be a bit careful, though. What value(s) of $t$ in the original domain will cause issues in the resulting expression?