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Chapter 2: Problem 45

Suppose that \(f\) is an odd function of \(x .\) Does knowing that \(\lim _{x \rightarrow 0^{+}} f(x)=3\) tell you anything about \(\lim _{x \rightarrow 0} f(x) ?\) Give rea- sons for your answer.

Short Answer

Expert verified
The two-sided limit does not exist because the one-sided limits are not equal.

Step by step solution

01

Understanding the Function Type

First, let's recall that an odd function has the property that for all values of \( x \), \( f(-x) = -f(x) \). This property will be important in analyzing the behavior of the function as \( x \) approaches zero.
02

Analyze Given Limit

We are given that \( \lim_{x \to 0^{+}} f(x) = 3 \). This means that as \( x \) approaches zero from the right (positive direction), the function \( f(x) \) approaches 3.
03

Determine the Left-Hand Limit

To find \( \lim_{x \to 0^{-}} f(x) \), use the property of odd functions. Since \( f(x) \) is odd, \( f(-x) = -f(x) \). Thus, \( \lim_{x \to 0^{-}} f(x) = \lim_{x \to 0^{+}} f(-x) \). Given the properties of limits, \( \lim_{x \to 0^{+}} (f(-x)) = -3 \).
04

Conclude the Two-Sided Limit

For the two-sided limit \( \lim_{x \to 0} f(x) \) to exist, both one-sided limits (from positive and negative) must be equal. However, \( \lim_{x \to 0^{+}} f(x) = 3 \) and \( \lim_{x \to 0^{-}} f(x) = -3 \). Since 3 is not equal to -3, these one-sided limits are not equal, hence \( \lim_{x \to 0} f(x) \) does not exist.

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Key Concepts

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

Odd Functions
An odd function is a unique type of function characterized by its symmetry. The mathematical property defining odd functions is expressed as \( f(-x) = -f(x) \) for all values of \( x \). This implies that if you take any \( x \) value and find its corresponding \( f(x) \), switching the sign of \( x \) (to get \( -x \)) will yield \( -f(x) \).
This property of odd functions indicates a type of rotational symmetry around the origin of the coordinate system.
  • If the point \((x, y)\) is on the graph, then \((-x, -y)\) is also on the graph.
  • Common examples of odd functions include \( f(x) = x^3 \) and \( f(x) = \sin(x) \).
This understanding of symmetry helps when determining limits, as knowing one side of a limit can be used to determine the behavior on the opposite side.
Two-Sided Limits
A two-sided limit is the value that a function approaches as the input approaches a specific value from both the left and the right. For a two-sided limit to exist at a point \( c \), the left-hand limit \( \lim_{x \to c^{-}} f(x) \) and the right-hand limit \( \lim_{x \to c^{+}} f(x) \) must be equal.
In mathematical terms,
  • \( \lim_{x \to c} f(x) \) exists if \( \lim_{x \to c^{-}} f(x) = \lim_{x \to c^{+}} f(x) \).
  • If these one-sided limits differ, the two-sided limit does not exist.
Understanding two-sided limits is important for comprehending how functions behave near a point and is essential for calculus concepts such as continuity and differentiability.
Behavior of Functions at a Point
The behavior of functions at a point focuses on how functions 'act' as the input \( x \) approaches a specific value. Several behaviors can occur:
  • A function can approach a finite number.
  • It can tend towards infinity, or it may not settle towards any particular value at all.
Knowing the behavior at a point is key in understanding limits and continuity.
For example, in the original exercise, the function \( f(x) \) approaches a specific value from the right \((3)\) and an altered value from the left \((-3)\), showcasing differing behaviors based on the approach direction.
This divergence highlights that while analyzing functions, one must consider the direction of approach to fully grasp a function's behavior at that point.

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Most popular questions from this chapter

At what points are the functions in Exercises 13-30 continuous? $$y=\frac{1}{|x|+1}-\frac{x^{2}}{2}$$

For what value of \(a\) is $$f(x)=\left\\{\begin{array}{ll}{a^{2} x-2 a,} & {x \geq 2} \\ {12,} & {x<2}\end{array}\right.$$ continuous at every \(x ?\)

You will find a graphing calculator useful. Let $$h(x)=\left(x^{2}-2 x-3\right) /\left(x^{2}-4 x+3\right)$$ a. Make a table of the values of \(h\) at \(x=2.9,2.99,2.999,\) and so on. Then estimate lim_{x} \rightarrow 3 \(h(x)\) . What estimate do you arrive at if you evaluate \(h\) at \(x=3.1,3.01,3.001, \ldots\) instead? b. Support your conclusions in part (a) by graphing \(h\) near \(x_{0}=3\) and using Zoom and Trace to estimate \(y\) -values on the graph as \(x \rightarrow 3\) . c. Find lim_{x\rightarrow3} \(h(x)\) algebraically.

At what points are the functions in Exercises 13-30 continuous? $$y=(2 x-1)^{1 / 3}$$

Find the limits in Exercises \(80-86\). $$\lim _{x \rightarrow+\infty}\left(2 x+\sqrt{4 x^{2}+3 x-2}\right)$$
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