Find the Limit:
I multiplied the numerator and denominator by the denominator, then started factoring things out and I ended up getting 1/64.
WebAssign is telling me this is wrong, but I can't figure out why. Any help?
Find the Limit:
I multiplied the numerator and denominator by the denominator, then started factoring things out and I ended up getting 1/64.
WebAssign is telling me this is wrong, but I can't figure out why. Any help?
Things can get tricky with the $\displaystyle -\infty$. To avoid trouble, make the substitution t = -x. Then you want
$\displaystyle \lim_{t \rightarrow +\infty} (-t + \sqrt{t^2 - 2t}) = \lim_{t \rightarrow +\infty} (\sqrt{t^2 - 2t} - t)$.
Note that:
$\displaystyle \displaystyle \frac{(\sqrt{t^2 - 2t} - t)(\sqrt{t^2 - 2t} + t)}{\sqrt{t^2 - 2t} + t} = \frac{(t^2 - 2t) - t^2}{\sqrt{t^2 - 2t} + t} = \frac{-2t}{\sqrt{t^2 - 2t} + t} = \frac{-2}{\sqrt{1 - \frac{2}{t}} + 1}$.
$\displaystyle \lim_{t \rightarrow +\infty} \frac{-2}{\sqrt{1 - \frac{2}{t}} + 1} = \frac{-2}{1 + 1} = -1$.