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Chapter 1: Problem 63

In the context of finding limits, discuss what is meant by two functions that agree at all but one point.

Short Answer

Expert verified
Two functions that 'agree at all but one point' share the same values for all inputs, except possibly at one point. In terms of limits in calculus, even if the behavior at that single exception point is undefined or irregular, the limit can be evaluated using the other function's values. The limit of the function that does not 'agree' at that point (if it exists) is the same as the limit of the other function.

Step by step solution

01

Understanding 'agreeing' functions

Two functions are said to 'agree' at a point if they yield the same result or value at that point. If we have two functions \(f(x)\) and \(g(x)\), they 'agree' at all points except at \(x = c\) if \(f(x) = g(x)\) for all \(x\) except possibly at \(x = c\). In such a case, it is said that \(f\) and \(g\) agree everywhere but possibly at \(c\).
02

Understanding Limits

A limit is a fundamental concept in calculus that describes the behavior of a function at a certain point. Specifically, the limit of a function \(f(x)\) as \(x\) approaches a certain value \(c\) is what the values of \(f(x)\) get closer to as \(x\) gets infinitely close to \(c\). This is written as \(\lim_{x \to c} f(x) = L\), where \(L\) is the limit. If the limit exists and is equal at a certain point for two functions that agree at all but one point, then the limit at that point can be evaluated using either function.
03

Applying the concept to agreeing functions

The principle of agreeing functions becomes especially important when dealing with limits. If two functions \(f\) and \(g\) agree at all points in an interval around \(c\) except possibly at \(c\) itself, and if \(\lim_{x \to c} f(x) = L\) exists, then \(\lim_{x \to c} g(x) = L\) as well. This means we can 'borrow' the function values from \(f\) to evaluate the limit of \(g\) at \(c\), even if \(g(c)\) is undefined or irregular. This is a key concept used in many limit calculations.

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