Question

(a) Prove that if \(\lim _{x \rightarrow c}|f(x)|=0,\) then \(\lim _{x \rightarrow c} f(x)=0\). (Note: This is the converse of Exercise \(74 .)\) (b) Prove that if \(\lim _{x \rightarrow c} f(x)=L,\) then \(\lim _{x \rightarrow c}|f(x)|=|L|\). [Hint: Use the inequality \(\|f(x)|-| L\| \leq|f(x)-L| .]\)

Short answer

The limit of the absolute value of a function is zero if the limit of the function itself is zero. And the absolute value of the limit of a function is the limit of the absolute value of the function.

Step-by-step solution

Proof of Part (a)

Assume that \(\lim _{x \rightarrow c}|f(x)|=0\). To prove \(\lim _{x \rightarrow c} f(x)=0\), we need to use the definition of a limit. That is, for every \(\epsilon >0\), there exists a \(\delta >0\) such that if \(0<|x-c|<\delta\), then \(|f(x)-0|<\epsilon\). But since \(\lim _{x \rightarrow c}|f(x)|=0\), it implies that \(|f(x)-0|=|f(x)|\) is less than any positive number \(\epsilon\), therefore, by virtue of the definition of limit, \(\lim _{x \rightarrow c} f(x)=0\).

Proof of Part (b)

Assume that \(\lim _{x \rightarrow c} f(x)=L\). To prove \(\lim _{x \rightarrow c}|f(x)|=|L|\), we can also resort to the definition of a limit. But in this case, making use of the inequality \(\|f(x)|-| L\| \leq|f(x)-L|\), for every \(\epsilon >0\), there exists a \(\delta >0\) such that if \(0<|x-c|<\delta\), then \(||f(x)|-|L||=|\|f(x)|-| L\| \leq|f(x)-L|<\epsilon\). Since as per the assumption, \(\lim _{x \rightarrow c}f(x)=L\), therefore \(|f(x)-L|<\epsilon\) which implies that \(||f(x)|-|L|| < \epsilon\) and this confirms that \(\lim _{x \rightarrow c}|f(x)|=|L|\).

Key concepts

Limit Properties

Understanding the properties of limits is crucial when working with functions, particularly as they approach a specific point. Imagine standing at the edge of a pond, throwing stones at a target; as you fine-tune your throw, you get closer and closer to hitting the target. Similarly, as a function approaches a particular point, the value of the function gets closer to the limit value.

When dealing with the absolute value of a function, which measures only the magnitude without considering the direction, this gets a bit more nuanced. In our exercise, we find that if the limit of the absolute value of a function as it approaches a point is zero, then the limit of the function itself is also zero. This stems from a fundamental limit property: if a function becomes arbitrarily close to a value (here zero), then its absolute value will also be arbitrarily close to the absolute value of that number (zero in this case).

This aligns with the limit properties that allow us to link the behavior of the function and its absolute value. These properties, like the distributive or associative properties in algebra, are tools that make it easier for us to manipulate and understand limits in different scenarios.

Absolute Value

Consider the absolute value as a type of 'distance' from zero along the number line, irrespective of direction. This metric is always non-negative since distance cannot be negative. In the context of limits, the absolute value plays a pivotal role because it allows us to discuss the nearness to a value independent of whether we approach from the left (negative side) or right (positive side).

The \textbf{triangle inequality} \(||f(x)|-|L|| \leq |f(x) - L|\) used in the solution utilizes the concept of absolute value to bound the difference between the absolute values of the function and the limit. This inequality is like saying that the shortest distance between two points on a number line is the direct path, without detours. By applying the triangle inequality to the limit process, we can effectively manage inequalities involving absolute values and make precise statements about the distance from the function to the limit.

Epsilon-Delta Definition of Limit

The epsilon-delta definition is the formal definition of a limit, and it's like a game of 'getting as close as we want'. In this game, \(\epsilon\) represents how close we want to get to the target (the limit's value), and \(\delta\) represents how close we need to be to the point of approach (the value that \(x\) is approaching) to achieve that.

To say that the limit of \(f(x)\) as \(x\) approaches \(c\) is \(L\) means that for every \(\epsilon > 0\) (no matter how small), we can find a \(\delta > 0\) such that when \(x\) is within \(\delta\) of \(c\) (excluding \(c\) itself), the value of \(f(x)\) is within \(\epsilon\) of \(L\). This precise definition allows us to prove limits formally, as demonstrated in the provided exercise, thereby bridging the intuitive notion of limits with rigorous mathematics.