2 TB - Calculus

Question

Limits: If f is a function of a single variable, what is the intuitive interpretation of $\underset{x\to a}{\lim }f\left(x\right)$? What is the formal definition? 

Step-by-Step Solution

Answer

The intuitive approach is if x approaches a point a, the value of the function at that point approaches a real number.

Let $f\left(x\right)$ be defined for all $x\ne a$ over an open interval containing a. Let L be a real number then $\underset{X\to a}{\lim }f\left(x\right)=L$ if for every $\epsilon >0$, there exists a $\delta >0$, such that if $0<|x-a|<\delta , then \left|f\right(x)-L|<\epsilon$.$f\left(x\right)$

Step 1: Given information

We need to determine what is the intuitive interpretation of $\underset{x\to a}{\lim }f\left(x\right)$ and what is the formal definition?

Step 2: Definition of limit.

Let we have given a function f(x). Then the intuitive approach is if the value of x approaches a point a, the value of the function at that point approaches a real number.

The Formal definition is: Let f(x) be defined for all $x\ne a$ over an open interval containing a. Let L be a real number then $\underset{X\to a}{\lim }f\left(x\right)=L$ if for every $\epsilon >0$, there exists a $\delta >0$, such that if $0<|x-a|<\delta , then \left|f\right(x)-L|<\epsilon$.