Page 282 - Calculus Workbook For Dummies

266 Part V: The Part of Tens

Fifth 3 over the “l”: 3 cases

where a limit fails to exist

The three cases are
At a vertical asymptote. This is an infinite discontinuity.
At a jump discontinuity.
With the limit at infinity or negative infinity of an oscillating function like lim cos x
x " 3
where the function keeps oscillating up and down forever, never honing in on a
single y-value.

Second 3 over the “i”: 3 parts to
the definition of continuity

First notice the oh-so-clever fact that the letter i can’t be drawn without taking your
pen off the paper and thus that it’s not continuous. This will help you remember that
the second and fourth 3s concern continuity.

The three-part, formal definition of continuity is in Chapter 3. The mnemonic will help
you remember that it’s got three parts. And — just like with the definition of a limit —
that’s enough to help you remember what the three parts are.

Fourth 3 over the “i”: 3 cases where

continuity fails to exist

The three cases are

A removable discontinuity — the highfalutin calculus term for a hole.
An infinite discontinuity.
A jump discontinuity.

Third 3 over the “m”: 3 cases where
a derivative fails to exist

Note that m often stands for slope, right? And the slope is the same thing as the
derivative. The three cases where it fails are

At any type of discontinuity.
At a cusp: a sharp point or corner along a function (this only occurs in weird
functions).
At a vertical tangent. (A vertical line has an undefined slope and thus an undefined
derivative.)