if B Approaches infinity, how he solved this

**moonnightingale** · February 15th 2011, 08:35 AM

If B approaches to infinity, how the book solved this
plz explain

**Ackbeet** · February 15th 2011, 08:52 AM

$\displaystyle\lim_{B\to\infty}C=\lim_{B\to\infty}B \,\log_{2}\left(1+\frac{S}{N_{0}B}\right)=\lim_{B\ to\infty}\frac{\ln\left(1+\frac{S}{N_{0}B}\right)} {(1/B)\ln(2)}.$

At this point, you employ L'Hopital's Rule by taking the derivative of the top w.r.t. B, and the derivative of the bottom w.r.t. B (NOT the quotient rule):

$\displaystyle \lim_{B\to\infty}\frac{\ln\left(1+\frac{S}{N_{0}B} \right)}{(1/B)\ln(2)}=\lim_{B\to\infty}\frac{\left(1/\left(1+\frac{S}{N_{0}B}\right)\right)\left(-\frac{S}{N_{0}B^{2}}\right)}{-(1/B^{2})\ln(2)}.$

Can you finish?

**topsquark** · February 15th 2011, 08:54 AM

Bah! You beat me to it, Ackbeet!

-Dan

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