Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/41

Rh Here we see the following facts pictured:

(a) For $$x=1$$, $$\log_ex=log_e1=0$$.

(b) For $$x>1$$, $$\log_ex$$ is positive and increases as $$x$$ increases.

(c) For $$1>x>0$$, $$\log_ex$$ is negative and increases in numerical value as $$x$$ diminishes, that is, $$\underset{x=0}{\operatorname{limit}}\;\log x=-\infty$$.

(d) For $$x\eqslantless0$$, $$log_ex$$ is not defined; hence the entire graph lies to the right of $$OY$$.

(5) Consider the function $$\frac{1}{x}$$, and set $$y=\frac{1}{x}.$$

If the graph of this function be plotted, it will be seen that as $$x$$ approaches the value zero from the left (negatively), the points of the curve ultimately drop down an infinitely great distance, and as $$x$$ approaches the value zero from the right, the curve extends upward infinitely far.

The curve then does not form a continuous branch from one side to the other of the axis of $$Y$$, showing graphically that the function is discontinuous for $$x=0$$, but continuous for all other values of $$x$$.

(6) From the graph of $$y=\frac{2x}{1-x^2}$$it is seen that the function $$\frac{2x}{1-x^2}$$is discontinuous for the two values $$x=\pm1$$, but continuous for all other values of $$x$$.

(7) The graph of $$y=\tan x$$shows that the function $$\tan x$$ is discontinuous for infinitely many values of the independent variable $$x$$, namely, $$x=\frac{n\pi}{2}$$, where $$n$$ denotes any odd positive or negative integer.

(8) The function $$\operatorname{arc~tan}x$$