Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/45

Rh {| \scriptstyle{\text{(1)}~}& \scriptstyle{\underset{x=0}{\operatorname{limit}}\;\frac{c}{x}}& \scriptstyle{=}& \scriptstyle{\infty;}& \scriptstyle{\frac{c}{0}}& \scriptstyle{=}& \scriptstyle{\infty.}\\ \scriptstyle{\text{(2)}~}& \scriptstyle{\underset{x=\infty}{\operatorname{limit}}\;cx}& \scriptstyle{=}& \scriptstyle{\infty;}& \scriptstyle{c\cdot\infty}& \scriptstyle{=}& \scriptstyle{\infty.}\\ \scriptstyle{\text{(3)}~}& \scriptstyle{\underset{x=\infty}{\operatorname{limit}}\;\frac{x}{c}}& \scriptstyle{=}& \scriptstyle{\infty;}& \scriptstyle{\frac{\infty}{c}}& \scriptstyle{=}& \scriptstyle{\infty.}\\ \scriptstyle{\text{(4)}~}& \scriptstyle{\underset{x=\infty}{\operatorname{limit}}\;\frac{c}{x}}& \scriptstyle{=}& \scriptstyle{0;}& \scriptstyle{\frac{c}{\infty}}& \scriptstyle{=}& \scriptstyle{0.}\\ \scriptstyle{\text{(5)}~}& \scriptstyle{\underset{x=-\infty}{\operatorname{limit}}\;a^x}& \scriptstyle{=}& \scriptstyle{+\infty,\text{ when }a<1;}& \scriptstyle{a^{-\infty}}& \scriptstyle{=}& \scriptstyle{+\infty.}\\ \scriptstyle{\text{(6)}~}& \scriptstyle{\underset{x=+\infty}{\operatorname{limit}}\;a^x}& \scriptstyle{=}& \scriptstyle{0,\quad\text{ when }a<1;}& \scriptstyle{a^{+\infty}}& \scriptstyle{=}& \scriptstyle{0.}\\ \scriptstyle{\text{(7)}~}& \scriptstyle{\underset{x=-\infty}{\operatorname{limit}}\;x^x}& \scriptstyle{=}& \scriptstyle{0,\quad\text{ when }a>1;}& \scriptstyle{a^{-\infty}}& \scriptstyle{=}& \scriptstyle{0.}\\ \scriptstyle{\text{(8)}~}& \scriptstyle{\underset{x=+\infty}{\operatorname{limit}}\;a^x}& \scriptstyle{=}& \scriptstyle{+\infty\text{ when}a>1;}& \scriptstyle{a^{+\infty}}& \scriptstyle{=}& \scriptstyle{+\infty.}\\ \scriptstyle{\text{(9)}~}& \scriptstyle{\underset{x=0}{\operatorname{limit}}\;\log_ax}& \scriptstyle{=}& \scriptstyle{+\infty\text{ when }a<1;}& \scriptstyle{\log_a0}& \scriptstyle{=}& \scriptstyle{+\infty.}\\ \scriptstyle{\text{(10)}~}& \scriptstyle{\underset{x=+\infty}{\operatorname{limit}}\;\log_ax}& \scriptstyle{=}& \scriptstyle{-\infty,\text{ when}a<1;}& \scriptstyle{\log_a(+\infty)}& \scriptstyle{=}& \scriptstyle{-\infty.}\\ \scriptstyle{\text{(11)}~}& \scriptstyle{\underset{x=0}{\operatorname{limit}}\;\log_ax}& \scriptstyle{=}& \scriptstyle{-\infty,\text{ when }a>1;}& \scriptstyle{\log_a0}& \scriptstyle{=}& \scriptstyle{-\infty.}\\ \scriptstyle{\text{(12)}~}& \scriptstyle{\underset{x=+\infty}{\operatorname{limit}}\;\log_ax}& \scriptstyle{=}& \scriptstyle{+\infty,\text{ when }a>1;}& \scriptstyle{\log_a(+\infty)}& \scriptstyle{=}& \scriptstyle{+\infty.} \end{array}$$
 * Written in the form of limits.|||||Abbreviated form often used.
 * $$\begin{array}{rrclrcl}
 * $$\begin{array}{rrclrcl}
 * }

The expressions in the second column are not to be considered as expressing numerical equalities ($$\scriptstyle{\infty}$$ not being a number); they are merely symbolical equations implying the relations indicated in the first column, and should be so understood.

Let $$\scriptstyle{O}$$ be the center of a circle whose radius is unity.

Let $$\scriptstyle{\operatorname{arc~}AM=\operatorname{arc~}AM'=x}$$, and let $$\scriptstyle{MT}$$ and $$\scriptstyle{M'T}$$ be tangents drawn to the circle at $$\scriptstyle{M}$$ and $$\scriptstyle{M'}$$. From Geometry,

Dividing through by $$\scriptstyle{2\sin x}$$, we get