Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

Prove that \(\lim _{x \rightarrow c} f(x)=0 \Leftrightarrow \lim _{x \rightarrow c}|f(x)|=0\).

Step-by-Step Solution

Answer

The limit of a function is 0 if and only if the limit of its absolute value is 0.

Step 1: Understanding the Exercise

We need to prove the equivalence between two statements involving limits: that the limit of a function is zero if and only if the limit of the absolute value of that function is zero. We will use the definition of limits to achieve this.

Step 1: Prove that if \(\lim_{x \to c} f(x) = 0\), then \(\lim_{x \to c} |f(x)| = 0\)

Assume \(\lim_{x \to c} f(x) = 0\). This means for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - c| < \delta\), then \(|f(x) - 0| < \epsilon\). Since \(|f(x)| = |f(x) - 0|\), the condition \(|f(x)| < \epsilon\) holds whenever \(0 < |x - c| < \delta\). Therefore, \(\lim_{x \to c} |f(x)| = 0\).

Step 2: Prove that if \(\lim_{x \to c}|f(x)| = 0\), then \(\lim_{x \to c} f(x) = 0\)

Assume \(\lim_{x \to c} |f(x)| = 0\). This means for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - c| < \delta\), then \(|f(x)| < \epsilon\). Since \(\left| |f(x)| - 0 \right| = |f(x)|\), the same condition \(|f(x)| < \epsilon\) directly implies \(|f(x) - 0| < \epsilon\). Therefore, \(\lim_{x \to c} f(x) = 0\).

Step 4: Concluding the Proof

The results from Step 1 and Step 2 show that both statements \(\lim_{x \to c} f(x) = 0\) and \(\lim_{x \to c} |f(x)| = 0\) imply each other. Therefore, we have proven that \(\lim _{x \rightarrow c} f(x)=0 \Leftrightarrow \lim _{x \rightarrow c}|f(x)|=0\).

Key Concepts

Absolute Value Function

The absolute value function is a crucial concept in many areas of mathematics, including calculus. It measures the distance of a number from zero on the real number line, without considering its direction. This means that the absolute value of both positive and negative numbers is always non-negative.
For instance, the absolute value of -3 is 3, symbolized as \(|-3| = 3\). This property makes the absolute value function particularly useful when dealing with limits and inequalities.
In calculus, the absolute value function can simplify expressions and solutions involving both positive and negative behaviors of a function as it approaches a particular point. Ultimately, understanding the role of absolute value helps in tackling limit problems, such as proving that \(\lim_{x \to c} f(x) = 0\) if and only if \(\lim_{x \to c} |f(x)| = 0\).

• The function is defined as \(|x| = x\) if \(x \geq 0\), and \(|x| = -x\) if \(x < 0\).
• It plays a vital role in understanding convergence toward zero by ensuring that whether a function swings above or below a given line, its absolute value captures only its magnitude.

Epsilon-Delta Definition of Limits

The epsilon-delta definition is a rigorous way to understand limits in calculus. It involves two small parameters: epsilon \(\epsilon\), representing an error bound on the function, and delta \(\delta\), corresponding to an interval around the point of interest.
Using these, we say the limit of \(f(x)\) as \(x\) approaches \(c\) is \(L\) if for every \(\epsilon > 0\), there is some \(\delta > 0\) such that whenever \(0 < |x - c| < \delta\), we have \(|f(x) - L| < \epsilon\).
This formalism is key to proving the equivalence between \(\lim_{x \to c} f(x) = 0\) and \(\lim_{x \to c} |f(x)| = 0\).

• This approach is an essential tool for justifying why a small change in \(x\) translates to a small change in \(f(x)\), within specified bounds.
• In our exercise, using the epsilon-delta definition allowed proving directionality, confirming both direct and reverse implications of limits involving absolute values.

Continuity in Calculus

Continuity is a fundamental concept in calculus, describing how functions behave smoothly and predictably.
A function is said to be continuous at a point \(c\) if the limit of the function as \(x\) approaches \(c\) is equal to the function's value at \(c\). More formally, \(\lim_{x \to c} f(x) = f(c)\).
This property is vital for understanding how function limits behave across their domain.
In proving assertions about limits, such as our exercise where \(\lim_{x \to c} f(x) = 0\) implies \(\lim_{x \to c} |f(x)| = 0\), continuity ensures that there are no jumps or breaks at the point \(c\).

• Continuity ensures a predictable flow of \(f(x)\) without unexpected deviations or discontinuities.
• It serves as a guarantee that limit statements are meaningful and calculable at and around specific points.

Understanding continuity helps in establishing conditions under which limits are consistent in different contexts, including those involving absolute values.