Evaluating the Limit of a Function: A Practical Example with ( f(x) 3x^2 )

Introduction

This article explains how to evaluate the limit of a function, specifically for $f(x) 3x^2$, as (x) approaches 2. We will use both direct substitution and the epsilon-delta definition to prove the correctness of our answer. This example is useful for understanding the fundamental concepts of limits in calculus.

Direct Substitution and Limit Laws

What is Direct Substitution?
Direct substitution is a method where you replace the variable (x) with the value it is approaching, in this case, 2, to find the limit of a function. This method is straightforward and applies to functions such as polynomials and rational functions, provided the function does not have a discontinuity at the point of interest.

Direct Substitution of ( f(x) 3x^2 )
To evaluate the limit as ( x ) approaches 2 for the function ( f(x) 3x^2 ), we can use direct substitution:

[lim_limits{x to 2} 3x^2 3(2)^2 3(4) 12]

Alternatively, we can break the expression into simpler parts using the properties of limits:

[lim_limits{x to 2} 3x^2 left( 3 lim_limits{x to 2} x right) cdot lim_limits{x to 2} 2 3 cdot 2 cdot 2 12]

Epsilon-Delta Definition of a Limit

What is the Epsilon-Delta Definition?
The epsilon-delta definition is a more rigorous way to understand the concept of a limit. It states that for any ( epsilon > 0 ), there exists a ( delta > 0 ) such that if ( 0

Applying the Epsilon-Delta Definition to ( f(x) 3x^2 )
We want to prove that:

[lim_limits{x to 2} 3x^2 12]

Starting with the inequality:

[|3x^2 - 12|

By factoring and using algebraic manipulation:

[|3x^2 - 12| 3|x^2 - 4| 3|x - 2||x 2|]

Let's assume ( |x - 2| [1

Also, ( 3 [|x 2|

Substituting back:

[3|x - 2||x 2|

Thus, we need:

[15|x - 2|

Solving for ( delta ):

[|x - 2|

Choose ( delta frac{epsilon}{15} ), then:

[0

Graphical and Numerical Verification

Graphical Representation
The behavior of the function ( f(x) 3x^2 ) as ( x ) approaches 2 is evident from the graph. As ( x ) approaches 2 from both sides, the function values approach 12.

Numerical Approximation
We can also verify the limit using numerical values:

x3x^2 1.98.03 1.998.003 1.9998.0003 212 2.00112.0003 2.0112.03 2.112.3

As the ( x ) values get closer to 2, the function values get closer to 12, confirming our analytical result.

Conclusion

In conclusion, we have evaluated the limit of the function ( f(x) 3x^2 ) as ( x ) approaches 2 using direct substitution and the epsilon-delta definition. Both methods yield the same result, which is 12. This example demonstrates the importance of understanding both practical and theoretical approaches to calculus.