Chapter 4 - Section 4.4 - Indeterminate Forms and l''Hospital''s Rule - 4.4 Exercises - Page 313: 82

a) $$ \begin{aligned} \lim _{R \rightarrow r^{+}} v &=\lim _{R \rightarrow r^{+}}\left[-c\left(\frac{r}{R}\right)^{2} \ln \left(\frac{r}{R}\right)\right] \\ &=-c r^{2} \lim _{R \rightarrow n^{+}}\left[\left(\frac{1}{R}\right)^{2} \ln \left(\frac{r}{R}\right)\right] \\ &=-c r^{2} \cdot \frac{1}{r^{2}} \cdot \ln 1\\ &=-c \cdot 0\\ &=0 \end{aligned} $$ As the insulation of a metal cable becomes thinner, the velocity of an electrical impulse in the cable approaches zero. b) $$ \begin{aligned} \lim _{r \rightarrow 0^{+}} v &=\lim _{r \rightarrow 0^{+}}\left[-c\left(\frac{r}{R}\right)^{2} \ln \left(\frac{r}{R}\right)\right]\\ &=-\frac{c}{R^{2}} \lim _{r \rightarrow 0^{+}}\left[r^{2} \ln \left(\frac{r}{R}\right)\right]\\ & \,\,\,\,\,[ \text{form is} \,\, 0 · \infty]\\ &=-\frac{c}{R^{2}} \lim _{r \rightarrow 0^{+}} \frac{\ln \left(\frac{r}{R}\right)}{\frac{1}{r^{2}}}\\ & \,\,\,\,\,[ \text{form is} \,\, \infty /\infty \,\, \text {and by using L'Hôpital's rule we have} ]\\ &=-\frac{c}{R^{2}} \lim _{r \rightarrow 0^{+}} \frac{\frac{R}{r} \cdot \frac{1}{R}}{\frac{-2}{r^{2}}}\\ &=-\frac{c}{R^{2}} \lim _{r \rightarrow 0^{+}}\left(-\frac{r^{2}}{2}\right)\\ &=0 \end{aligned} $$ As the radius of the metal cable approaches zero, the velocity of an electrical impulse in the cable approaches zero.

a) $$ \begin{aligned} \lim _{R \rightarrow r^{+}} v &=\lim _{R \rightarrow r^{+}}\left[-c\left(\frac{r}{R}\right)^{2} \ln \left(\frac{r}{R}\right)\right] \\ &=-c r^{2} \lim _{R \rightarrow n^{+}}\left[\left(\frac{1}{R}\right)^{2} \ln \left(\frac{r}{R}\right)\right] \\ &=-c r^{2} \cdot \frac{1}{r^{2}} \cdot \ln 1\\ &=-c \cdot 0\\ &=0 \end{aligned} $$ As the insulation of a metal cable becomes thinner, the velocity of an electrical impulse in the cable approaches zero. b) $$ \begin{aligned} \lim _{r \rightarrow 0^{+}} v &=\lim _{r \rightarrow 0^{+}}\left[-c\left(\frac{r}{R}\right)^{2} \ln \left(\frac{r}{R}\right)\right]\\ &=-\frac{c}{R^{2}} \lim _{r \rightarrow 0^{+}}\left[r^{2} \ln \left(\frac{r}{R}\right)\right]\\ & \,\,\,\,\,[ \text{form is} \,\, 0 · \infty]\\ &=-\frac{c}{R^{2}} \lim _{r \rightarrow 0^{+}} \frac{\ln \left(\frac{r}{R}\right)}{\frac{1}{r^{2}}}\\ & \,\,\,\,\,[ \text{form is} \,\, \infty /\infty \,\, \text {and by using L'Hôpital's rule we have} ]\\ &=-\frac{c}{R^{2}} \lim _{r \rightarrow 0^{+}} \frac{\frac{R}{r} \cdot \frac{1}{R}}{\frac{-2}{r^{2}}}\\ &=-\frac{c}{R^{2}} \lim _{r \rightarrow 0^{+}}\left(-\frac{r^{2}}{2}\right)\\ &=0 \end{aligned} $$ As the radius of the metal cable approaches zero, the velocity of an electrical impulse in the cable approaches zero.