---- ## General information #### *Definition* - A **limit** describes the **output of a function as its input *approaches* a given value**. - Limits are primarily used to solve problems that involve **infinitely large or small values** which cant be expressed as a finite term. #### *Properties* - Limits can be used to find the characteristics of a functions **asymptotes** since they are places where the function's output or input **approaches infinity**. - Limits can be used to find what an input's corresponding solution **would be** in cases where the function would **otherwise be unsolvable or result in a "rouge value".** These are **singular, lone** inputs that result in an output that breaks the functions continuity. *(Example: x = 17 on the graph below)* - Limits can be evaluated from the negative or positive side and described below. **If no direction is given the limit must approach the *same value* from both directions in order to exist!** >[!tip] >$\color{Bisque}\text{\small negative notation:}\space\Large\lim_{x\rightarrow 1^-}\space\space\space\text{\small positive notation:}\space\lim_{x\rightarrow 1^+}$ >$\textsf{\large A\color{Bisque} negative \color{white}limit indicates you are approaching x from the\color{Bisque} left}$ >$\textsf{\large A\color{Bisque} positive \color{white}limit indicates you should approaching x from the\color{Bisque} right}$ >[!example] >##### Visualizing limits >- **At -7** the overall limit of this function is **negative infinity** since that's what it approaches from both the negative and positive direction. >- **At 5** the overall limit of this function **does not exist** since it **approaches two different values**. Approaching five from the negative side results in a limit equal to 4 whereas the positive side would equal 6. >- **At 13** the **limit equals 2** since both the negative and positive directions converge towards that value. >- **At 17** the **limit equals 5** even though there is a **"hole"** for the same reason as 13. > >![[Limitsd.png|850]] --- ## Notation - When evaluating a limit it is extremely important to use the correct notation, without it your work is technically incorrect. Furthermore, this notation must be used **every time you re-write the equation no matter how it has been algebraically modified**.In fact, It is only when all variables have been replaced with numerical values that it is no longer needed. - Correct limit notation is shown below. **lim** must be writen **before the function** and below it you must indicate what **value *(b)* the variable *(a)* is approaching.** >[!quote] >$\text{\large\color{LightCyan} Correct limit notation}$ >$\color{white}\Large\lim_{a\rightarrow b}\space\Big[ f(a)\Big]$ >--- >$\text{\large\color{LightCyan}Example}$ >$\color{white}\left(\lim_{x\rightarrow 1}\frac{x^2-1}{x-1}\right) =\left(\lim_{x\rightarrow 1}\frac{(x+1)\cancel{(x-1)}}{\cancel{(x-1)}}\right)=\Big(x+1\Big)=\color{LightCyan} 2$ --- ## Limit laws >[!quote] >| ***Name*** | ***Rule*** | >|-|-| >| **Direct substitution property** | $\lim_{x\rightarrow a}f(x)=\color{LightCyan}\Large f(a)$ >| **Sum/difference rule** | $\lim_{x\rightarrow a}\Big[f(x)\pm g(x)\Big]\rightarrow\color{LightCyan}\lim_{x\rightarrow a}f(x)\pm\lim_{x\rightarrow a}g(x)$ >| **Constant rule** | $\lim_{x\rightarrow a}\Big[c\cdot f(x)\Big]\rightarrow \color{LightCyan}c\cdot\lim_{x\rightarrow a}f(x)$ >| **Product rule** | $\lim_{x\rightarrow a}\Big[ f(x)\cdot g(x)\Big]\rightarrow\color{LightCyan}\lim_{x\rightarrow a}f(x)\cdot\lim_{x\rightarrow a}g(x)$ >| **Quotient rule** | $\lim_{x\rightarrow a}\left[\frac{f(x)}{g(x)}\right]\rightarrow\color{LightCyan}\frac{\lim_{x\rightarrow a}f(x)}{\lim_{x\rightarrow a}g(x)}$ >| **Power rule** | $\lim_{x\rightarrow a}\Big[f(x)^n\Big]\rightarrow\color{LightCyan}\left[\lim_{x\rightarrow a}f(x)\right]^n$ --- ## Finite Limits - To evaluate a finite limit, **use the direct substitution property and solve the resulting equation.** If this equation is **undefined** $\frac{1}{0}$ or **indeterminate** $\frac{0}{0}$ the initial limit likely needs to be **algebraically "re-arranged"** in a way that allows you to solve the resulting limit - If using the direct substitution property **results in a defined/solvable term don't re-arrange the function!** If you do the answer will be **incorrect!** *This is why its always important to attempt the DSP before doing ANYTHING ELSE.* >[!tip] >#### Re-arranging unsolvable finite limits >- The following are a few techniques that are commonly used to make undefined/indeterminate limits solvable. They primarily apply to fractional limits which represent most unsolvable limits. >- An **undefined limit** is a fractional limit that has a **zero in its denominator only**. An **indeterminate limit has a zero in *both* the denominator and numerator**. >$\newline$ >$\text{\Large\color{Bisque}Factoring}$ >$\text{\small Without factoring: }\left(\lim_{x\rightarrow 1}\frac{x^2-1}{x-1}\right)\rightarrow\left(\frac{1^2-1}{1-1}\right)=\color{Bisque}\Big(\space\frac{0}{0}\space\text{ undefined}\space\Big)$ >$\text{\small With factoring: }\left(\lim_{x\rightarrow 1}\frac{x^2-1}{x-1}\right) \rightarrow\left(\lim_{x\rightarrow 1}\frac{(x+1)\cancel{(x-1)}}{\cancel{(x-1)}}\right)\rightarrow\Big(x+1\Big)=\color{bisque} 2$ >$\newline$ >$\text{\Large\color{Bisque}Conjugation}$ >$\text{\small Without conjugation: }\left(\lim_{x\rightarrow 4}\frac{2-\sqrt{x}}{4-x}\right)\rightarrow\left(\frac{2-\sqrt{4}}{4-4}\right)=\color{Bisque}\Big(\space\frac{0}{0}\space\text{ undefined}\space\Big)$ >$\text{\small With conjugation: }\lim{x\rightarrow 4}\left(\frac{2-\sqrt{x}}{4-x}\right)\cdot\left(\frac{2+\sqrt{x}}{2+\sqrt{x}}\right)$ #### *Table of values* - Another way you can solve finite limits is by using something called **a table of values**. This approach is often a last resort since it relies on brute force instead of algebraic techniques. - Even though most teachers wont let you use this technique it **can be useful for verifying your answer**. $\newline$ $\color{LightCyan}\small\text{Input values that approach the value specified in the limit}$ $\color{LightCyan}\small\text{If the limit was }\lim_{x\rightarrow 0}f(x)\text{ we would record the outputs of:}$ $\color{LightCyan}f(0.1),\space f(0.01,\space f(0.001)\space \text{ and }\space f(-0.1),\space f(-0.01),\space f(-0.001)$ $\newline$ - Using the outputted values we can create a **table of values**. This table allows us to determine what value the function is *approaching*. Unless the limit specifies a direction you should use values **from both sides *(positive and negative)***. If these two *"sides"* approach **different values then you know the limit does not exist.** - It is standard practice to record the output of three values from both sides. These values should approach that specified in the limit. --- ## Non-finite limits - When evaluating a limit that involves an infinity **many of the traditional techniques outlined above wont work.** Since there is no formulaic, outlined way to solve infinite limits you will have to rely on primarily on your intuition. - Treat the infinity symbol **like a variable** when algebraically manipulating it. - Non-finite limits that **do not contain a fraction are very easy to solve** since the solution can only be one of three things: **zero, infinity or negative infinity**. As a result this section primarily focuses on fractional, non finite limits. $\newline$ $\color{white}\text{Basic properties of non-finite limits}$ $\color{grey}\small\text{The right arrow symbol }(\rightarrow)\text{ means "approaching" and n is any non-zero real number}$ $\newline$ $\color{white}\text{Zero and n: }\Biggl[\color{LightCyan}\left(\frac{\rightarrow n}{\rightarrow 0}\right)\rightarrow \space\infty\color{white}\Biggl]\color{LightCyan}\space\space\space\color{white}\Biggl[\color{LightCyan}\left(\frac{\rightarrow 0}{\rightarrow n}\right)\rightarrow0\color{white}\Biggl]\color{LightCyan}\space\space\space\color{white}\Biggl[\color{LightCyan}\left(\frac{\rightarrow n}{\rightarrow n}\right)\rightarrow 1\color{white}\Biggl]$ $\color{white}\text{Infinity and zero: }\Biggl[\color{LightCyan}\left(\frac{\rightarrow\infty}{\rightarrow0}\right)\rightarrow\infty\color{white}\Biggl]\space\space\space\color{white}\Biggl[\color{LightCyan}\left(\frac{\rightarrow0}{\rightarrow\infty}\right)\rightarrow 0\color{white}\Biggl]\color{LightCyan}\space\space\space\color{white}\Biggl[\color{LightCyan}\left(\frac{\rightarrow\infty}{\rightarrow\infty}\right)\rightarrow 1\color{white}\Biggl]\color{LightCyan}$ $\color{white}\text{Infinity and n: }\Biggl[\color{LightCyan}\left(\frac{\rightarrow n}{\rightarrow\infty}\right)\rightarrow0\color{white}\Biggl]\color{LightCyan}\space\space\space\color{white}\Biggl[\color{LightCyan}\left(\frac{\rightarrow\infty}{\rightarrow n}\right)\rightarrow\infty\color{white}\Biggl]\space\space\space\color{white}\text{Inequalities: }\Big[\color{LightCyan}\infty<\infty^2\color{white}\Big]\space\space\space\Big[\color{LightCyan}\space n\cdot\infty=\infty\color{white}\Big]$ $\newline$ >[!tip] >##### *Division technique* >- This technique makes it much easier to tell **which infinities are larger than others** and which terms need to be crossed out *(have no effect on the result).* The method involves dividing **every term by $x$ to some power $n$**. Usually $n$ is equal to the value of the **largest exponent** in the original equation *(on x of course)*. > *Still to add: squeeze theorem trig limits * --- ## Calculator >[!cite] > >--- ><iframe src="https://www.wolframalpha.com/calculators/limit-calculator" width="850" height="1200"></iframe>