1.46 Examples of limits

Strategy: First simplify expressions, then take limits.

Problem: Find the limits as $R\rightarrow\infty$ of

Solution. (i) $\log(2R+1)-\log(R+2)=\,{\log\left(\frac{2R+1}{R+2}\right)}{=\log\left(\frac{2+\frac{1}{R}}{1+\frac{2}{R}}\right).}$

Since $\frac{2+\frac{1}{R}}{1+\frac{2}{R}}\rightarrow\frac{2}{1}=2$ as $R\rightarrow\infty$, we get: $\lim_{R\rightarrow\infty}f(R)=\log 2$.

(ii) $2\log(R+1)-\log(3R^{2}+1)=\,{\log\left(\frac{(R+1)^{2}}{3R^{2}+1}\right)=}\,{\log\left(\frac{(1+\frac{1}{R})^{2}}{3+\frac{1}{R^{2}}}\right).}$

As $\frac{(1+\frac{1}{R})^{2}}{3+\frac{1}{R^{2}}}\rightarrow\frac{1}{3}$ as $R\rightarrow\infty$, we obtain: $\lim_{R\rightarrow\infty}g(R)=-\log 3$.