What is it?
By definition, limits are related to functions and graphs, and it describes the behavior of a function as it approaches a given value.
Notation and definition
Given defined around , except at itself, the limit of when tends to is , denoted by the formula below: In other words, if we have a function where a given number makes it undefined, it’s established a limit on the given number. Take for example the function below:
The function is always 1 despite any values inputted as , but when , the function becomes undefined, as does not have a definition. Because of that, any other value is accepted, except for the value of 1. We can get infinitely close to 1, as long as it never actually reaches it, the function is still defined.
Rewriting it so it matches the limit notation presented up above, it would be:
\scriptsize\text{"The limit of\unicode{0x2026}"}&\qquad\scriptsize\text{"\unicode{0x2026}the function }f\text{\unicode{0x2026}"} \\ \searrow\qquad&\qquad\swarrow \\ \LARGE\displaystyle\lim_{x\to 1}&\LARGE f(x) = 1 \\ \nearrow\qquad \\ \scriptsize\text{"\unicode{0x2026}as }x\text{ approaches }1\text{."} \end{aligned}$$ ___ # Approaching infinitesimally closer Limits are about a **range of numbers** that **get closer and closer** to a specified value, but ***never quite reaches it***. Let's analyze another function: $$f(x) = x^{2}, \, x \neq 2 $$ In this case, the **function is undefined** at *$x = 2$*, so its established **a limit** on this point. Let's rewrite the function taking the limit into account:\lim_{{x \to 2}} , x ^ {2} \text{, \qquad resulting in the graph below:}
![[Limits Parabola.png]] As *$x$* approaches *2* from **either side**, *$f(x)$* also gets closer and closer to **4**. This happens to both sides **infinitely**. ## A limit must be the same from both sides For example, $f(x) = \frac{1}{x - 2}$, is defined for all real numbers **except** at *$x = 2$* . In this case, it would create a [[Asymptote]] at $x = 2$, where the values would get ***infinitesimally closer and closer*** but never **reaching 2**. However, this would not classify as a limit, ***because it's not possible to approach the same value from both sides.*** ![[Limits Asymptote.png]] > *When approaching from the left, $y \to -\infty$ and from the right, $y \to +\infty$.* ___ # Solving a limit Let's solve the following equation:\lim _{x \rightarrow 1}\left[\frac{2 x^2-3}{3 x+1}\right] = \quad ?
It asks for the limit of the function *— inside the square brackets —*. We can solve the function using $x = 1$ to discover its image *— **y.***\lim _{x \rightarrow 1}=\left[\frac{2 x^2-3}{3 x+1}\right]=\frac{2 \cdot(1)^2-3}{3 \cdot 1+1}=-\frac{1}{4}
It means that when $x \to 1$, the **function will result in** $y = -1/4$.