# Some algebraic manipulation.

#### ensbana

I've just got back to study some subjects that require maths, and my skill is pretty rusty. Could anyone help me with putting bounds on the following expression?

For $$\displaystyle k$$ a positive integer, and $$\displaystyle \delta \in (0,1)$$: $$\displaystyle \frac{\sqrt{(1- \delta)^{2k} + 4\delta} \ - (1-\delta)^k}{2\delta}$$

#### romsek

Math Team
Well it's bounded below by 1. I suspect it increases without bound as $k\to \infty$

ensbana

#### idontknow

$\lim_{\delta \rightarrow 1 } \frac{\sqrt{(1- \delta)^{2k} + 4\delta} \ - (1-\delta)^k}{2\delta} =0$.

$\lim_{\delta \rightarrow 0 } \frac{\sqrt{(1- \delta)^{2k} + 4\delta} \ - (1-\delta)^k}{2\delta} =\lim_{\delta \rightarrow 0 } \dfrac{4\delta }{4\delta }=1$.

ensbana

#### romsek

Math Team
$\lim_{\delta \rightarrow 1 } \frac{\sqrt{(1- \delta)^{2k} + 4\delta} \ - (1-\delta)^k}{2\delta} =0$.

$\lim_{\delta \rightarrow 0 } \frac{\sqrt{(1- \delta)^{2k} + 4\delta} \ - (1-\delta)^k}{2\delta} =\lim_{\delta \rightarrow 0 } =\lim_{\delta \rightarrow 0 } \dfrac{4\delta }{2\delta }=2$.
Mathematica returns a limit of 1 as $\delta \to 0$