Limits in Calculus

The Role of Function Values at a Point in Determining Limits

A pivotal concept in calculus is that the limit of a function as x approaches a certain value is not affected by the function's value at that point. To illustrate, consider a function that coincides with f(x) = x for all x except at x = 1, where it has a different value or is undefined. The limit as x approaches 1 is unaffected and remains L = 1. This is because limits are concerned with the values that f(x) approaches as x nears 1, rather than the actual value of f(x) when x is exactly 1. This principle is crucial, especially when dealing with functions that have discontinuities or points of undefined behavior.

Practical Application of the Limit Definition

Applying the formal definition of a limit to a specific function involves a systematic approach. Begin by conjecturing a limit value L. For a given ε, find a δ that ensures all function values fall within an ε-distance from L whenever x is within a δ-distance from the point of interest. This may require analyzing the function's graph closely or computing a table of values to gauge the proximity to L. If necessary, calculate two δ values for the left and right approaches and use the smaller one to meet the definition's criteria. The proof is completed by demonstrating that for any ε > 0, a corresponding δ can be found such that 0 < |x - a| < δ guarantees |f(x) - L| < ε.

Illustrative Examples and Essential Insights in Limit Calculations

Consider the constant function f(x) = k. The limit of f(x) as x approaches any value 'a' is simply k, as the function's value is constant. For a linear function like f(x) = 2x - 3, the limit as x approaches 7 is 11. Establishing the inequality |2x - 14| < ε and solving for δ allows us to find a suitable δ that satisfies the limit definition, often chosen to be ε/2 or smaller. These examples highlight that the choice of δ depends on the specific function, the target point 'a', and the chosen ε. The fundamental insight is that the limit of a function as x approaches 'a' is L if, for every ε > 0, there exists a δ > 0 such that 0 < |x - a| < δ implies |f(x) - L| < ε. This relationship is denoted by lim(x→a) f(x) = L, signifying that as x approaches 'a' indefinitely close, the function value converges to L.