Question

Prove that if \(\lim _{x \rightarrow c} f(x)\) exists and \(\lim _{x \rightarrow c}[f(x)+g(x)]\) does not exist, then \(\lim _{x \rightarrow c} g(x)\) does not exist.

Short answer

If the limit of a function as \(x\) approaches a specific value exists but the limit of the sum of this function and another function as \(x\) approaches the same value doesn't exist, then it is concluded that the limit of the other function as \(x\) approaches this value doesn't exist as well.

Step-by-step solution

Formulate the problem using the properties of limits

The assumption is that \( \lim_ {x \rightarrow c} f(x) \) exists. Let's denote this limit by \(L\). Written formally, \( \lim_ {x \rightarrow c} f(x) = L \). If we knew that \( \lim _{x \rightarrow c} g(x) \) also existed and denoted this limit by \(M\), then from the properties of limits, it would follow that \( \lim _{x \rightarrow c} [f(x) + g(x)] = L + M \). However, this contradicts the given that \( \lim _{x \rightarrow c} [f(x) + g(x)] \) does not exist.

Conclude based on contradiction

Since we've arrived at a contradiction, we can conclude that our assumption that \( \lim _{x \rightarrow c} g(x) \) exists must be false. Therefore, we have proof by contradiction that if \( \lim _{x \rightarrow c} f(x)\) exists and \( \lim _{x \rightarrow c} [f(x) + g(x)] \) does not exist, then \( \lim _{x \rightarrow c} g(x) \) does not exist.

Key concepts

Proof by Contradiction

Understanding the concept of proof by contradiction can be pivotal in grasping more complex mathematical arguments. In essence, the strategy involves assuming the exact opposite of what you're trying to prove and showing that this assumption leads to an illogical conclusion, a contradiction. This means that the original statement must be true. For example, suppose we are trying to establish that a specific limit does not exist, as in our exercise. By assuming the contrary, that the limit does exist, and demonstrating that this leads to a situation that cannot possibly be true given the other conditions of the problem, we can confidently assert that our initial statement was correct.

This logical process not only solidifies understanding but also plays a crucial role in advancing mathematical theory. It's like a detective realizing that their suspect couldn't have committed the crime because otherwise, an impossible scenario would occur. That's why in our original exercise, when we assume the limit of g(x) as it approaches c exists, and this leads directly to a contradiction with given conditions, we're forced to conclude that the limit of g(x) does indeed not exist.

Properties of Limits

Now let's focus on the properties of limits, which are the rules that limits follow. These properties help us to manipulate and combine limits in predictable ways. One of the basic properties is that if the limits of two functions exist as x approaches c, then the limit of their sum or difference also exists and is equal to the sum or difference of their limits respectively. What's more, multiplication by a scalar and multiplication or division of functions also follow similar rules, provided the limits exist and the denominator isn't zero.

In our exercise, we relied on this property when we stated that if the limits of f(x) and g(x) as x approaches c both existed, then the limit of the sum f(x) + g(x) would simply be the sum of those two limits. Since the exercise set forth that the limit of the sum did not exist, this conflicted with what the properties of limits would dictate, should the limit of g(x) exist alongside the known limit of f(x). Here, these properties act as the foundation upon which our contradiction is constructed.

Nonexistent Limits

The final concept we tackle is the notion of nonexistent limits. When we say a limit does not exist, we're essentially stating that there is no single value that the function's output gets arbitrarily close to as the input approaches some value c. This could happen for several reasons - the function might oscillate without approaching any value, grow without bound, or even take on different values depending on the direction from which c is approached (known as a 'multi-directional' limit).

In the context of our exercise, the failure of the sum f(x) + g(x) to settle as x approaches c indicates 'unsettling' behavior by at least one of the component functions. Given that f(x) has a limit, this 'unsettling' must originate from g(x). Such insights into the behavior of functions are critical in calculus for understanding continuity, integrals, and other advanced topics. Detecting nonexistent limits often requires us to patiently analyse a function's behavior, often employing a mix of graphical intuition and algebraic skill.