I'm trying to solve a limit problem but I've never encountered the one like this before.
$$ \lim_{t\to 0}\left(\frac{1}{t\sqrt{1+t}}-\frac{1}{t}\right)$$
I multiply the right side by $\frac{t\sqrt{1+t}}{t\sqrt{1+t}}$ and combine the terms to get:
$$ \lim_{t\to 0}\left(\frac{1 - t\sqrt{1+t}}{t\sqrt{1+t}}\right)$$
I can factor out the $t\sqrt{1+t}$ and I am left with:
$$1 - 1 = 0$$
This seems wrong to me. I can't explain why, maybe I just feel that getting a zero after all that work seems lame. Am I following the right steps?