math:limits - wiki.epix.icu

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====== Limits ======
A limit is the y-value of a function as it approaches a certain value. As a function arbitrarily approaches the corresponding x-value of a limit, the function output also gets arbitrarily closer to the limit itself. Note that limits do not necessarily mean that the value is a discontinuity, there are both continuous and non-continuous limits. Limits do not necessarily have to exist on the function.

Types of limits
  * $\frac{n_1}{n_2}$ || $\frac{0}{n}$ are both valid
  * $\frac{n}{0}$ is undefined
  * $\frac{0}{0}$ is indeterminate

===== Indeterminate =====
If a rational function, when plugging in a discontinuity((roots of the denominator)), evaluates to $\frac{0}{0}$, it is called an indeterminate. To solve for the discontinuity, one must simplify away the denominator which causes the discontinuity((as you can't divide by zero)). When finding limits of indeterminates, use [[math:polynomial_division|Polynomial Division]]((both long and synthetic work)) or factoring to simplify. Any factors removed are called removable discontinuities((holes on the graph)). You can use the simplified function to find the limit.

===== Undefined =====
When evaluating a function that results in a divide by zero, specifically $\frac{1}{0}$, it's result is undefined. These types of functions are easy to tell as they clearly approach infinity when graphed. However, infinity is not a number, it is a concept, As such, the limit for this function __**Does Not Exist**__.

===== Piecewise =====
{{:math:2020-11-09-225246_291x216_scrot.png |}} With a function like this, where both sides approach different values, the limit also __**Does Not Exist**__. Both sides must approach the same y-value for a limit to exist.

===== lim x->inf =====
As x approaches $\infty$ or $-\infty$, x will approach the ratio of the leading coefficents((coefficent from the term of highest degree)).
Trace:
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