Q. 3 - Calculus

Question

Let $y = f\left(x\right)$. What is the definition of $\underset{x\to c}{\lim }f\left(x\right) = L$?

Step-by-Step Solution

Answer

As such an $L$does not exist, we can say $\underset{x\to c}{\lim }f\left(x\right)$ does not exist.

Step 1. Given Information

$y=f\left(x\right)$ is a function

We need to define$\underset{x\to c}{\lim }f\left(x\right)=L$

Step 2. Defining lim x → c f ( x ) = L

The definition is as such
for $g>0$, there exists $\delta >0$ such that $\left|f\right(x)-L|<\epsilon$ whenever $x\in E$
(where $E\subseteq R$, the set of real numbers is R) and $0<|x-c|<\delta$
In this case, we can say that the limit of the function $f\left(x\right)$ as $x$ tends to $\text{'}c\text{'}$ exists and it is $\text{'}L\text{'}$ and we write it as

$\underset{x\to c}{\lim }f\left(x\right)=L$ or $f\left(x\right)\to L$ as $x\to c$