Calculus Wrap: August 22

Functions can have “holes” (breaks in continuity at one point)
Defined limits as numbers “getting arbitrarily closer to” L.
$\lim\limits_{x \to a}{f(x)} = L$ if $\lim\limits_{x \to a^-}{f(x)} = \lim\limits_{x \to a^+}{f(x)}$ and both exist.
- Using the terms “lefthand” and “righthand” limit, abbreviated to LH and RH.
- RH limit is “approaches from righthand” or “values of $a$ where $x \gt a$ gets arbitrarily close to $L$”
Vertical asymptotes happen if the function shoots up to $\infty$ or $-\infty$
- In other words, $\exists a$ where $\lim\limits_{x\to a^{\pm}}{f(x)} = \pm\infty $ ($\pm$ is the LH or RH limit, my notation)
Infinite limits
Identifying vertical asymptotes and limits with graphs

$f(a) \not\equiv \lim\limits_{x \to a}$
May not have confidence that the LH limit and RH limit are sometimes equal and sometimes not. Might second guess for no reason.
Might miss vertical asymptotes.
For functions such as $\frac{x}{x}$ students may simplify it to $1$ not understanding the “hole” at $x=0$
confusing $a$ with $L$ in the form $\lim\limits_{x \to a}{f(x)} = L$

$\frac{x^2-4}{x-2}$ reduces to $x+2$, but only when $x \ne 2$ to prevent a $\frac{0}{0}$ error
$\lim\limits_{x \to 0}{\frac{x}{|x|}} DNE $
$f(x) = |x|$ is piecewise and $\lim\limits_{x\to 0}{|x|}$ DNE
piecewise graphs