Special limiting values

The following examples are of special importance in the study of the Calculus. In the following examples $a > 0$ and $c \ne 0$.

Eqn number	Written in the form of limits	Abbreviated form often used
(1)	$\lim_{x \to 0} \frac{c}{x} = \infty$	$\frac{c}{0} = \infty$
(2)	$\lim_{x \to \infty} cx = \infty$	$c \cdot \infty = \infty$
(3)	$\lim_{x \to \infty} \frac{x}{c} = \infty$	$\frac{\infty}{c} = \infty$
(4)	$\lim_{x \to \infty} \frac{c}{x} = 0$	$\frac{c}{\infty} = 0$
(5)	$\lim_{x \to -\infty} a^x, = +\infty$ , when $a < 1$	$a^{-\infty} = +\infty$
(6)	$\lim_{x \to +\infty} a^x = 0$, when $a < 1$	$a^{+\infty} = 0$
(7)	$\lim_{x \to -\infty} a^x = 0$, when $a > 1$	$a^{-\infty} = 0$
(8)	$\lim_{x \to +\infty} a^x = +\infty$, when $a > 1$	$a^{+\infty} = +\infty$
(9)	$\lim_{x \to 0} \log_a\ x = +\infty$, when $a < 1$	$\log_a\ 0 = +\infty$
(10)	$\lim_{x \to +\infty} \log_a\ x = -\infty$, when $a < 1$	$\log_a(+\infty) = -\infty$
(11)	$\lim_{x \to 0} \log_a\ x = -\infty$, when $a > 1$	$\log_a\ 0 = -\infty$
(12)	$\lim_{x \to +\infty} \log_a\ x = +\infty$, when $a > 1$	$\log_a(+\infty) = +\infty$

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