Understanding the Limit and Continuity of a Function at a Point

Understanding the fundamental concepts of limits and continuity is crucial in calculus and analysis. This article explores how to find the limit of a specific function and determine its continuity. We will focus on the behavior of the function ( f(x) x^2 ) as ( x ) approaches 0, utilizing sequences of rational and irrational numbers, and the Squeeze Theorem.

Introduction

Consider a function ( f ) defined as ( f(x) x^2 ) for any real number ( x ). We aim to investigate the limit of ( f(x) ) as ( x ) approaches a point ( a ) in the real numbers ( mathbb{R} ). Specifically, we will set ( a 0 ) to explore the behavior of the function at this point.

Limits and Sequences

One of the key methods in analyzing limits involves the use of sequences. According to the density property of the rational and irrational numbers, for any real number ( a ), we can find sequences of rational numbers ( {x_i} ) and irrational numbers ( {y_i} ) that both converge to ( a ).

For the function ( f(x) x^2 ), let's examine the sequences ( {x_i} ) and ( {y_i} ) that converge to 0:

The sequence ( {x_i} ) consists of rational numbers, and since ( x_i ) approaches 0, ( f(x_i) x_i^2 ) also approaches 0. The sequence ( {y_i} ) consists of irrational numbers, and since ( y_i ) approaches 0, ( f(y_i) y_i^2 ) also approaches 0.

If the limit ( lim_{x to a} f(x) ) exists, then for the sequences ( {x_i} ) and ( {y_i} ), we must have:

( lim_{i to infty} f(x_i) lim_{i to infty} f(y_i) )

This implies:

( a^2 0 )

Given that the square of any non-zero real number is non-zero, the only solution for ( a ) is 0. Therefore, the limit ( lim_{x to a} f(x) ) exists if and only if ( a 0 ).

Proving the Limit and Continuity at ( x 0 )

Now let's prove that the limit ( lim_{x to 0} f(x) ) exists and equals 0. We know that for any real number ( x ), ( 0 leq f(x) leq x^2 ). This can be rewritten as:

( 0 leq x^2 leq x^2 )

We observe that:

( lim_{x to 0} 0 0 ) ( lim_{x to 0} x^2 0 )

By the Squeeze Theorem, if a function ( g(x) ) is sandwiched between two functions ( h(x) ) and ( k(x) ) that both converge to the same limit ( L ), then ( g(x) ) also converges to ( L ). Here, ( 0 leq f(x) leq x^2 ) and both ( lim_{x to 0} 0 0 ) and ( lim_{x to 0} x^2 0 ), thus:

( lim_{x to 0} f(x) 0 )

Futhermore, since ( f(0) 0^2 0 ), the function ( f ) is continuous at ( x 0 ).

Summary

In summary, the limit ( lim_{x to a} f(x) ) exists if and only if ( a 0 ). The function ( f ) is continuous at ( x 0 ) because ( f(0) 0 ) and the limit as ( x ) approaches 0 equals 0.

Through the exploration of sequences and the Squeeze Theorem, we have shown that the function ( f(x) x^2 ) is continuous only at ( x 0 ).