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Chapter 2: Problem 194

In the following exercises, use the precise definition of limit to prove the given one-sided limits. \(\lim _{x \rightarrow 0^{+}} f(x)=-2,\) where \(f(x)=\left\\{\begin{array}{ll}8 x-3, & \text { if } x<0 \\ 4 x-2, & \text { if } x \geq 0\end{array}\right.\).

Short Answer

Expert verified
Using precise definitions, \( \delta = \frac{\epsilon}{4} \) proves \( \lim_{x \rightarrow 0^+} f(x) = -2 \).

Step by step solution

01

Understanding the Problem

We need to prove the one-sided limit as \( x \) approaches \( 0^+ \) for the function \( f(x) \). Specifically, we have \( f(x) = 4x - 2 \) when \( x \geq 0\). We need to show that \( \lim_{x \rightarrow 0^+} f(x) = -2 \).
02

Review the Definition of Limit

To prove this limit, we use the precise (epsilon-delta) definition of a limit. For \( \lim_{x \rightarrow 0^+} f(x) = -2 \), for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < x < \delta \), then \( |f(x) + 2| < \epsilon \).
03

Substituting the Function

For \( x \) approaching \( 0^+ \), we use \( f(x) = 4x - 2 \). Therefore, we need to show \( |(4x - 2) + 2| < \epsilon \). This simplifies to \( |4x| < \epsilon \).
04

Solve the Inequality

The inequality \( |4x| < \epsilon \) simplifies to \( 4|x| < \epsilon \). Since \( x \) is positive as it approaches 0 from the right, \( |x| = x \). So, \( 4x < \epsilon \).
05

Choose Delta

To satisfy the inequality \( 4x < \epsilon \), we can choose \( \delta = \frac{\epsilon}{4} \). Thus, if \( 0 < x < \frac{\epsilon}{4} \), then \( 4x < \epsilon \) is satisfied, proving the limit.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

One-Sided Limits
One-sided limits are limits where we approach a point from only one side鈥攅ither from the left (denoted as \( x \to a^- \)) or from the right (denoted as \( x \to a^+ \)). These are particularly useful when a function behaves differently depending on the side from which we approach. In our exercise, we wanted to find the right-hand limit of the function \( f(x) \) as \( x \) approaches 0 from the right.
  • In this case, the function \( f(x) = 4x - 2 \) was used for \( x \geq 0 \).
  • The aim was to verify that as \( x \) approaches 0 from the positive side, the function approaches -2.
One-sided limits help in understanding discontinuities and different behaviors of piecewise functions at a particular point. They are essential when checking the existence of a full limit at a point by ensuring both one-sided limits exist and are equal.
Epsilon-Delta Definition
The epsilon-delta definition of a limit is a formal way to describe the concept of limits. It specifies that for a function \( f(x) \), a limit \( L \), and a point \( a \), \( \lim_{x \to a} f(x) = L \) if:
  • For every positive number \( \epsilon \), there exists a positive number \( \delta \).
  • If \( 0 < |x - a| < \delta \), then \( |f(x) - L| < \epsilon \).
This definition is crucial because it quantifies how close \( f(x) \) gets to \( L \) as \( x \) approaches \( a \). In the exercise:
  • We showed this by proving that for any \( \epsilon > 0 \), there is a \( \delta \) such that if \( 0 < x < \delta \) then \( |f(x) + 2| < \epsilon \). Thus, it aligns with \( L = -2 \) as \( x \rightarrow 0^+ \).
This method helps in understanding the stability and behavior of functions near specific points.
Precise Definition of Limit
Using the precise definition of a limit means applying the epsilon-delta approach to prove a limit exactly, rather than estimating or calculating its value arbitrarily. This approach ensures mathematical rigor and eliminates ambiguity.
  • It involves understanding the target limiting value \( L \) and function behavior as \( x \) nears \( a \).
  • In our problem, the goal was to prove \( \lim_{x \to 0^+} f(x) = -2 \) using a structured process.
  • The precise definition is essential for confirming that any limit computation aligns with the theoretical foundation of calculus.
By following the steps outlined, including choosing a \( \delta \) value specifically derived as \( \delta = \frac{\epsilon}{4} \), the rigorous framework of the precise definition of limit ensures accuracy and reliability when dealing with limits in calculus.

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Most popular questions from this chapter

In the following exercises, write the appropriate \(\varepsilon-\delta\) definition for each of the given statements. $$\lim _{x \rightarrow c} h(x)=L$$

In the following exercises, find the value(s) of \(k\) that makes each function continuous over the given interval. $$ f(x)=\left\\{\begin{aligned} e^{k x}, & 0 \leq x<4 \\ x+3, & 4 \leq x \leq 8 \end{aligned}\right. $$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist. $$\lim _{x \rightarrow 1} \frac{x^{2}-1}{\sqrt{x}-1}$$

In the following exercises, use the precise definition of limit to prove the given infinite limits. $$\lim _{x \rightarrow-1} \frac{3}{(x+1)^{2}}=\infty$$

In the following exercises, use the precise definition of limit to prove the given infinite limits. $$\lim _{x \rightarrow 0} \frac{1}{x^{2}}=\infty$$
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