> [!example]- **Preface:** The Big Picture > Take a look at the following function: $f(x)=\frac{x-1}{x-1}$ > > At first glance you might say that it can be simplified to $f(x)=1$ since the numerator and denominator are equivalent. However, it's important to remember that any fraction with a denominator of $0$ (or any number divided by $0$ in general) is **undefined**. > > ![[Limits.png|212]] > > Because of this, we must say that $x=1$ is *outside the domain* of $f$. This is why it's so important to know the [[Domain and Range]] of a function. > > Limits are similar in this sense, as they can be used to determine what values cannot exist within functions and how the function behaves when approaching them. > - If the limit did not exist at this point, then we would know that the two sides the function approach from arrive at different values # Limits A **limit** describes how a function behaves **as it approaches** a point. > This page will focus on the **intuitive definition of a limit** > > [[The Precise Definition of a Limit]] is far more complex and one of the most challenging definitions early on in calculus When the y limit and x limit are the same you say that the The limit only exists if the left side limit is equal to the right side limit > [!summary] Formal Definition > Let $f(x)$ be a function defined for all values of $x$ in an open interval $I$ containing $a$, with the possible exception of $a$ itself. > > Let $L$ be a real number. > > If for all values of $f(x)$ as $x$ approaches $a$ is $L$, then we say that the limit of $f(x)$ as $x$ approaches $a$ is $L$ > $\lim_{x \to \alpha} f(x) = L$ ## Alternative Translations The notation $\lim_\limits{x \to a} f(x) = L$ can also be stated in the following ways: - As $x$ gets closer to $a$, $f(x)$ gets closer to $L$ - $f(x) \to L$ as $x \to a$ ## Even vs Odd Integer Limits ![[Asymptotes Example.png|500]] ## Evaluating Limits The first two **limit laws** are known as the **basic limit results**. They, along with the other limit laws listed further below, allow us to evaluate the limit of many types of **algebraic functions**. > Essentially you will use the following limit laws to simplify the equation to only consist of the two limit formats in the basic laws above. > [!information]+ Basic Limit Results > For any real number $a$ and any constant $c$, > $\begin{align} \text{i.} && \lim_{x \to a} x = a \\ \text{ii.} && \lim_{x \to a} c = c \end{align}$ ### The Main Limit Laws The following properties of limits are known as **the limit laws**: - Let $f(x)$ and $g(x)$ be defined for all $x \ne a$ over some open interval containing $a$. - Assume that $L$ and $M$ are real numbers such that $\lim_\limits{x \to a}f(x)=L$ and $\lim_\limits{x \to a}f(x)=M$. - Let $c$ be a constant. Then, each of the following statements hold: > [!summary]- The Main Limit Laws > *Sum/Difference Law:* > $\lim_\limits{x \to a}[f(x) \pm g(x)] = \lim_\limits{x \to a}f(x) \pm \lim_\limits{x \to a}g(x)$ > --- > *Constant Multiple Law:* > $\lim_\limits{x \to a}c \cdot f(x) = c \cdot \lim_\limits{x \to a}f(x)$ > $\lim_\limits{x \to a}c = c$ > > --- > *Product Law:* > $\lim_\limits{x \to a}[f(x) \cdot g(x)] = \lim_\limits{x \to a}f(x) \cdot \lim_\limits{x \to a}g(x)$ > > --- > *Quotient Law:* > $\lim_\limits{x \to a}\frac{f(x)}{g(x)} = \frac{\lim_\limits{x \to a}f(x)}{\lim_\limits{x \to a}g(x)} = \frac{L}{M} \text{ for } M \ne 0$ > > --- > *Power Law:* > $\lim_\limits{x \to a}(f(x))^n = [\lim_\limits{x \to a}f(x)]^n$ > - For every positive integer $n$ > --- > *Root Law for Limits:* > $\lim_\limits{x \to a}\sqrt[n]{f(x)} = \sqrt[n]{\lim_\limits{x \to a}f(x)}$ > - For all $L$ if $n$ is odd > - For $L \ge 0$ if $n$ is even and $f(x) \ge 0$ ### Indeterminate Forms > https://mathematicalmysteries.org/lhopitals-rule/ | Indeterminate Form | Explanation | | ------------------------- | ----------- | | $\frac{0}{0}$ | | | $\frac{\infty}{\infty}$ | | | $\infty-\infty$ | | | $0^0$ | | | $1^\infty$ | | | $\infty^0$ | | | $0\cdot \infty$ | | ### L’Hopital’s Rule $\lim_{x \to a} \frac{f(x)}{g(x)}=\lim_{x \to a} \frac{f'(x)}{g'(x)}=$ - We can only use this rule when the limit is an indeterminate form ## One-Sided Limits One-sided limits, otherwise known as left or right handed-limits Left Sided: - $\lim_{x\rightarrow c^-}f(x)$ - Approaching from the left (lower valued) domain Right Sided: - $\lim_{x\rightarrow c^+}f(x)$ - Approaching from the right (higher valued) domain ## Limits at Infinity ![[Types of Discontinuities.png|400]] > [!important] What is infinity? > It’s important to realize that infinity: > 1. Is not a number > 2. Is larger than any real number > 3. Represents unboundedness There are three types of infinite limits: - Infinite Limits From the Left - Infinite Limits From the Right - Two-Sided Infinite Limits ### Evaluating Limits at Infinity Personal Thoughts: - Evaluating limits involving infinity does definitely have a formal method, although I’ve found it easiest to just use basic intuition and think of things as “very large” or “very small” when determining the priority of what should remain in the function