Problem 23 - Calculus Early Transcendentals

Question

Use the precise definition of a limit to prove the following limits. $$\left.\lim _{x \rightarrow 0} x^{2}=0 \text { (Hint: Use the identity } \sqrt{x^{2}}=|x| .\right)$$

Step-by-Step Solution

Answer

Question: Prove that the limit of the function \(x^2\) as \(x\) approaches \(0\) is equal to \(0\), using the identity \(\sqrt{x^2} = |x|\) to express the function in terms of an absolute value. Answer: By expressing the function in terms of an absolute value and establishing the relationship between \(\delta\) and \(\epsilon\) as \(\delta = \epsilon\), we have proven that for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - 0| < \delta\), then \(|x^2 - 0| < \epsilon\). Therefore, the limit of the function \(x^2\) as \(x\) approaches \(0\) is equal to \(0\).

Step 1: Express the function in terms of absolute value

The first step is to rewrite the given function, \(x^2\), using the hint provided. We can take the square root of both sides of the equation \(x^2=y\) to get: $$\sqrt{x^2}= \sqrt{y}$$ Using the hint, we get: $$|x| = \sqrt{y}$$ Now, replace \(y\) with \(x^2\) and rewrite the equation as: $$|x| = \sqrt{x^2}$$

Step 2: Set up the precise definition of a limit

According to the precise definition of a limit, \(\lim_{x \rightarrow 0} x^2 = 0\) if for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that for all \(x\), if \(0<|x-0|<\delta\) then \(|x^2 - 0| < \epsilon\). We want to establish this definition for the given function, with the given hint. So, we want to find a value for \(\delta\) in terms of \(\epsilon\) such that it satisfies the definition.

Step 3: Use the function expressed in terms of absolute value

Using the expression we found in step 1, we can rewrite the definition \(|x^2 - 0| < \epsilon\) as: $$|\sqrt{x^2}| < \epsilon$$ Since we know that \(\sqrt{x^2} = |x|\), we can rewrite the inequality as: $$|x| < \epsilon$$

Step 4: Establish the relationship between delta and epsilon

We have established that if \(|x| < \epsilon\), then \(|x^2 - 0| < \epsilon\). Now we need to find a relationship between \(\delta\) and \(\epsilon\). Since we know that the inequality \(0 < |x - 0| < \delta\) must hold, we can replace the absolute value \(|x|\) with \(\delta\), giving us: $$|x| < \delta$$ Since we want to relate \(\delta\) and \(\epsilon\), we can set \(\delta = \epsilon\). This means that, for every \(\epsilon > 0\), there exists a \(\delta = \epsilon\) such that: $$0<|x|<\delta\Rightarrow|x|<\epsilon$$

Step 5: State the proof conclusion

We have found that the value of \(\delta\) that satisfies the precise definition of a limit is \(\delta = \epsilon\). This means that for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - 0| < \delta\), then \(|x^2 - 0| < \epsilon\). Thus, we have used the precise definition of a limit to prove that: $$\lim_{x \rightarrow 0} x^2 = 0$$

Key Concepts

Precise definition of limit

The precise definition of a limit is a cornerstone in calculus. It provides a rigorous way to determine what happens to a function as it approaches a specific point. Particularly, it states that the limit of a function \( f(x) \) as \( x \) approaches \( c \) is \( L \), if for every \( \epsilon > 0 \) (no matter how small), there exists a \( \delta > 0 \) such that whenever \( 0 < |x-c| < \delta \), then \( |f(x)-L| < \epsilon \).

This concept ensures that the function \( f(x) \) gets arbitrarily close to \( L \) as \( x \) gets close to \( c \), providing an exact way to define limits rather than relying on intuition alone. This definition guarantees that we can make the function's value as close as we want to \( L \) by restricting \( x \) to a sufficiently small interval around \( c \).

In the exercise, we used this idea to show that \( \lim_{x \to 0} x^2 = 0 \) by carefully choosing a \( \delta \) for every given \( \epsilon \), enabling the limit condition to be satisfied.

Delta-epsilon definition

The delta-epsilon definition is the heart of the precise definition of limits. It provides a structured approach to prove that a limit exists. Essentially, it says that for every small positive number \( \epsilon \) (epsilon), there must be a corresponding small positive number \( \delta \) (delta), which can bind the difference between \( x \) and \( c \) within \( \delta \) in such a way that the difference between \( f(x) \) and \( L \) stays within \( \epsilon \).

This method is essential for proving limits rigorously. Here's how it works:

• Start by choosing any small \( \epsilon > 0 \).
• Find a \( \delta > 0 \) such that if \( 0 < |x - c| < \delta \), then \( |f(x) - L| < \epsilon \).

This approach was demonstrated to prove that \( x^2 \) approaches 0 as \( x \) approaches 0. For this particular problem, we cleverly used \( \delta = \epsilon \) to meet the requirement of the definition, thereby proving the limit.

Square root identity

The square root identity, \( \sqrt{x^2} = |x| \), is a useful mathematical identity that was applied in this proof. Understanding and using this identity can simplify expressions and make solving limits easier.

This identity states that the square root of a squared number returns its absolute value. It helps in separating variables when working on proofs, such as the epsilon-delta proofs in calculus.

In our exercise, the hint suggested using this identity to express \( x^2 \) in terms of \(|x|\). Consequently, we utilized \( \sqrt{x^2} = |x| \) to simplify the absolute expression from \( |x^2 - 0| < \epsilon \) into \( |x| < \epsilon \).

This simplification was crucial in finding a suitable \( \delta \) and proving the limit by showing that \( |x| < \epsilon \) whenever \( x \) is close enough to zero, thus allowing the application of the precise definition of limit smoothly. The identity is a fundamental concept that helps students approach problems with more clarity and understanding.