Problem 75 - Calculus of a Single Variable

Question

Proof Prove that if the limit of \(f(x)\) as \(x\) approaches \(c\) exists, then the limit must be unique. [Hint: Let \(\lim _{x \rightarrow c} f(x)=L_{1}\) and \(\lim _{x \rightarrow c} f(x)=L_{2}\) and prove that \(L_{1}=L_{2} . ]\)

Step-by-Step Solution

Answer

If the limit of \(f(x)\) as \(x\) approaches \(c\) exists, then the limit must be unique. This is confirmed through a proof by contradiction.

Step 1: Assume Two Different Limits

Let's start off by assuming there are two different limits \(L_1\) and \(L_2\) such that \(\lim _{x \rightarrow c} f(x)=L_{1}\) and \(\lim _{x \rightarrow c} f(x)=L_{2}\). That is, for every \(\varepsilon > 0\), there exist \(\delta _1 > 0\) such that if \(0 < |x - c| < \delta _1\), then \(|f(x) - L_1| < \varepsilon\), and \(\delta _2 > 0\) such that if \(0 < |x - c| < \delta _2\), then \(|f(x) - L_2| < \varepsilon\).

Step 2: Obtain a Contradiction

Let's consider \(\delta = min\{\delta _1 , \delta _2\}\). Then for \(0 < |x - c| < \delta\), both the inequalities \(|f(x) - L_1| < \varepsilon\) and \(|f(x) - L_2| < \varepsilon\) hold. This implies \(|L_1 - L_2| = |L_1 - f(x) + f(x) - L_2| \leq |f(x) - L_1| + |f(x) - L_2| < \varepsilon + \varepsilon = 2 \varepsilon\). There exists a positive number \(2 \varepsilon\) such that \(|L_1 - L_2| < 2 \varepsilon\), which can only be possible if \(L_1 = L_2\).

Step 3: Conclude the Proof

We assumed that there were two different limits, \(L_1\) and \(L_2\), and we have now shown that this leads to the conclusion that \(L_1 = L_2\). This is a contradiction as we initially thought that \(L_1\) and \(L_2\) were different numbers. We can therefore conclude that if the limit of \(f(x)\) as \(x\) approaches \(c\) exists, then the limit must be unique.

Key Concepts

Uniqueness of Limits

The uniqueness of limits revolves around the idea that a function cannot approach two different values as the input gets arbitrarily close to a specific point. To understand this concept, imagine a graph where the curve seems to settle at a particular height above a given point on the x-axis. If the limit were not unique, the curve would impossibly settle at two heights at once, which is not feasible.
When we say \( \lim_{x \to c} f(x) = L \), it conveys the information that as \(x\) gets closer to \(c\) from any direction, \(f(x)\) behaves more and more like the value \(L\). If there were another limit \(L_2\), similar criteria would hold, and you would eventually realize that these criteria contradict the essence of having two different final destinations for \(f(x)\) at that point.
This leads to a crucial inference: if a limit exists, it has to be unique because otherwise, it creates a contradiction, which breaks mathematical logic and consistency.

Epsilon-Delta Definition of Limit

To rigorously establish what it means for a function \(f(x)\) to have a limit as \(x\) approaches a point \(c\), we use the epsilon-delta definition. This definition sets a formal ground:

• Given any \(\varepsilon > 0\) representing how close you want \(f(x)\) to get to the limit \(L\),
• There exists a \(\delta > 0\) such that whenever \(0 < |x - c| < \delta\), \(f(x)\) is within \(\varepsilon\) of \(L\), that means \(|f(x) - L| < \varepsilon\).

Think of \(\varepsilon\) as a tiny range around the proposed limit \(L\) that we wish \(f(x)\) to fall within, and \(\delta\) as the maximum allowable distance we'll let \(x\) stray from \(c\) to achieve this proximity.
This definition is vital because it gives a clear and testable condition for limits, free from ambiguity or intuition, ensuring mathematical calculations and proofs can be reliably made.

Proof by Contradiction

Proof by contradiction is an elegant method of reasoning where you assume the opposite of what you want to prove. For limits, assuming non-uniqueness shows how contradiction is a potent tool.
Imagine \(\lim_{x \to c} f(x) = L_1\) and \(\lim_{x \to c} f(x) = L_2\), with \(L_1 eq L_2\). You initially accept both values as valid limits. Using epsilon-delta conditions, both limits should hold true for small distances around \(c\): \(f(x)\) should close in around \(L_1\) and \(L_2\).
But then, mathematics demands that the distance between these supposed limits, \(|L_1 - L_2|\), must be smaller than any positive number, which is only possible if \(L_1 = L_2\). The logic collapses our initial assumption of two different limits, confirming that no two distinct limits can exist simultaneously for a single converging point of \(f(x)\).
Thus, contradiction has shown us that the uniqueness of limits is more than a logical rule; it is a necessary condition for the stability of mathematical structures.