Problem 112 - Calculus of a Single Variable

Question

Proof (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 \(110 .\) ) (b) Prove that if \(\lim _{x \rightarrow c} f(x)=L,\) then \(\lim _{x \rightarrow c}|f(x)|=|L|\) \([\text {Hint} : \text { Use the inequality }\|f(x)|-| L\| \leq|f(x)-L| .]\)

Step-by-Step Solution

Answer

Both statements have been proved. The limits \(\lim _{x \rightarrow c} f(x)\) and \(\lim _{x \rightarrow c}|f(x)|\) are indeed connected as proposed in the exercise.

Step 1: Proof of Statement (a)

We need to prove that if \(\lim _{x \rightarrow c}|f(x)|=0,\) then \(\lim _{x \rightarrow c} f(x)=0\). Proof: Suppose \(\lim _{x \rightarrow c}|f(x)|=0\), that means for all \(\epsilon > 0,\) there exist a \(\delta_{1} > 0\) such that whenever \(0< |x - c| < \delta_{1}\), \(|f(x)| < \epsilon\). However, since \(|-f(x)| = |f(x)|\), we can also find a \(\delta_{2}\) so that \(|-f(x)| < \epsilon\) for all \(x\) in the domain of \(f\) for which \(0< |x - c| < \delta_{2}\). Combining these, the condition \(0< |x - c|< \min{(\delta_{1}, \delta_{2})}\) implies that \(-\epsilon < f(x) < \epsilon\). Therefore, applying the definition of a limit, \(\lim _{x \rightarrow c} f(x)=0\).

Step 2: Proof of Statement (b)

We need to prove that if \(\lim _{x \rightarrow c} f(x)=L\), then \(\lim _{x \rightarrow c}|f(x)|=|L|\). Proof: Suppose \(\lim _{x \rightarrow c} f(x)=L\), that means for all \(\epsilon > 0,\) there exist a \(\delta\) such that whenever \(0< |x - c| < \delta\), \(|f(x) - L| < \epsilon\). Using the inequality \(||f(x)|-|L||\leq|f(x) - L|\), we know that \(||f(x)| - |L|| < \epsilon\) whenever \(0< |x - c| < \delta\), which according to the definition of limit, means \(\lim _{x \rightarrow c}|f(x)|=|L|\).

Key Concepts

Absolute Value Properties

Understanding the properties of absolute values is crucial when dealing with limits in calculus. The absolute value function measures the distance of a number from zero on the real number line, regardless of direction. That is, it strips away any negative sign, leaving only the magnitude. For example, the absolute value of both \( -5 \) and \( 5 \) is \( 5 \) because both points are five units away from zero.

There are several key properties of absolute values that are helpful to know:

• \( |a| \geq 0 \) for all real numbers \( a \), because distances are non-negative.
• \( |ab| = |a||b| \) suggests that the absolute value of a product is the product of the absolute values.
• The triangle inequality: \( |a + b| \leq |a| + |b| \), which states that the absolute value of a sum is less than or equal to the sum of absolute values.

These properties become particularly useful when manipulating and proving statements about limits, as seen in the given exercise where the inequality \( ||f(x)| - |L|| \leq |f(x) - L| \) is applied to ultimately prove that \( \lim _{x \rightarrow c}|f(x)|=|L| \).

Epsilon-Delta Definition of Limit

The epsilon-delta definition of a limit forms the precise mathematical foundation for the concept of limits. This definition not only helps in formulating limits but also ensures rigorous proofs within calculus. It states that the limit of \( f(x) \) as \( x \) approaches \( c \) is \( L \) if for every \( \epsilon > 0 \) there exists a \( \delta > 0 \) such that whenever \( 0 < |x - c| < \delta \) it follows that \( |f(x) - L| < \epsilon \).

In simple terms, \( \epsilon \) represents how close we want \( f(x) \) to be to \( L \) and \( \delta \) determines the range around \( c \) within which \( f(x) \) will stay within the \( \epsilon \) distance from \( L \). This concept ensures that \( f(x) \) gets arbitrarily close to \( L \) as \( x \) approaches \( c \) and is the backbone for understanding and proving limits as seen in the step-by-step solution of the exercise.

Limit Proofs

Proving a limit involves showing that the function \( f(x) \) approaches a specific value as \( x \) gets infinitely close to a point \( c \). The proofs rely heavily on the epsilon-delta definition of limits and often require a logical sequence of steps that implies every condition set by this definition is met.

Structure of a Limit Proof

In a typical limit proof like those in the exercise examples, one starts by assuming the condition that needs to be established. Next, we choose an \( \epsilon \) and demonstrate the existence of a \( \delta \) that satisfies the limit's definition. This involves manipulating the inequalities, often utilizing the properties of absolute values and other mathematical techniques to isolate \( |f(x) - L| \) and show that it can be made less than \( \epsilon \) for all \( x \) within a \( \delta \) distance from \( c \).

When proving limits involving absolute values, as shown in the solution, it's common to employ the reverse triangle inequality \( ||a| - |b|| \leq |a - b| \) to directly relate to the epsilon-delta condition. The proofs in calculus are not merely about crunching numbers but understanding the underlying behaviors of functions and using precise logical arguments to demonstrate their properties.