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

Here we see the following facts pictured:

(a) For $x=1$, $\log _{e}x=log_{e}1=0$.

(b) For $x>1$, $\log _{e}x$ is positive and increases as $x$ increases.

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

(d) For $x\eqslantless 0$, $log_{e}x$ 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 $x=\pm 1$, 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

$$\operatorname {arc~tan} x$$