https://collinswyatt.com/calc/ Recent content in Calcs on Collins Hugo -- gohugo.io en-us https://collinswyatt.com/calc/aug22/ Mon, 01 Jan 0001 00:00:00 +0000 https://collinswyatt.com/calc/aug22/ Covered today: 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 Students may have trouble with: $f(a) \not\equiv \lim\limits_{x \to a}$ May not have confidence that the LH limit and RH limit are sometimes equal and sometimes not. https://collinswyatt.com/calc/aug24/ Mon, 01 Jan 0001 00:00:00 +0000 https://collinswyatt.com/calc/aug24/ Covered today: Functions used to define limits: Constant functions output the same constant regardless of input, eg. $f(x) = k$. Identity functions output the input, eg. $f(x) = x$ Properties of limits: $f(x)$ and $g(x)$ are functions where $\lim\limits_{x \to a}{f(x)} = L$ and $\lim\limits_{x \to a}{g(x)} = M$. Then the following properties are true: $$\lim\limits_{x \to a}{f(x) + g(x)} = L + M$$ $$\lim\limits_{x \to a}{f(x) * g(x)} = LM$$ $$\lim\limits_{x \to a}{\frac{f(x)}{g(x)}} = \frac{L}{M} \text{ if } M \neq 0$$ This is meant to teach us that you can break apart limits into multiple pieces, see example.