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Chapter 2: Problem 51

Define what it means to say that \(\lim _{x \rightarrow 0} g(x)=k\)

Short Answer

Expert verified
The limit \( \lim_{x \to 0} g(x) = k \) means \( g(x) \) approaches \( k \) as \( x \) approaches 0. It's formalized by the \( \epsilon-\delta \) definition.

Step by step solution

01

Understanding Limits

The notation \( \lim_{x \to a} g(x) = k \) indicates that as the variable \( x \) approaches the value \( a \), the function \( g(x) \) approaches the value \( k \). In simpler terms, \( g(x) \) gets closer and closer to \( k \) as \( x \) gets closer to \( a \).
02

Formal Definition

The formal definition of a limit \( \lim_{x \to a} g(x) = k \) is based on the \( \epsilon-\delta \) definition. This means: For every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( 0 < |x - a| < \delta \), it follows that \( |g(x) - k| < \epsilon \).
03

Breaking Down \( \epsilon-\delta \)

The \( \epsilon-\delta \) definition describes the precision of \( g(x) \) approaching \( k \). Where \( \epsilon \) represents the tolerance level for the limit, indicating how close \( g(x) \) should be to \( k \) to be considered close enough. The value \( \delta \) shows the window around \( x = a \) where \( x \) must lie for \( g(x) \) to stay within the \( \epsilon \) range of \( k \).
04

Application to Given Limit

In the specific context of \( \lim_{x \to 0} g(x) = k \), it means that as \( x \) gets arbitrarily close to 0, \( g(x) \) approaches the value \( k \). You can make \( g(x) \) as close as you like to \( k \) by choosing \( x \) sufficiently close to 0.

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Key Concepts

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

Epsilon-Delta Definition
The epsilon-delta definition of a limit is a critical concept in understanding limits in calculus. It provides a precise way to define what it means for a function to approach a certain value as the input gets close to a particular point. Here's how it works:

The key players in this definition are two Greek symbols:
  • \( \epsilon \) (epsilon) - This represents the desired level of precision or the margin by which we want the output of the function (\( g(x) \)) to be close to the limit \( k \).
  • \( \delta \) (delta) - This indicates how close \( x \) should be to the value \( a \) to ensure that \( g(x) \) is within the \( \epsilon \) range of \( k \).
For the definition to hold true, for each \( \epsilon > 0 \), there must exist a \( \delta > 0 \) such that whenever \( 0 < |x - a| < \delta \), it follows that \( |g(x) - k| < \epsilon \).

In simpler terms, this means we can make \( g(x) \) as close to \( k \) as we want by choosing \( x \) values close enough to \( a \).
Limit Approaching a Value
Understanding limits can initially seem a bit abstract, but breaking it down makes it more manageable. When we say \( \lim_{x \to a} g(x) = k \), we are essentially saying that the function \( g(x) \) gets closer to the value \( k \) as \( x \) approaches \( a \). This idea is fundamental in calculus as it describes the behavior of functions near a particular point.

To visualize this:
  • Imagine a point on the graph of \( g(x) \) that we are zeroing in on as \( x \) moves towards \( a \). The closer \( x \) gets to \( a \), the closer the output \( g(x) \) gets to our target \( k \).
  • It doesn't necessarily mean that \( g(x) \) will equal \( k \) when \( x = a \). In fact, \( \lim_{x \to a} g(x) \) does not even require \( g(a) \) to be defined.
This approach helps us understand and work with functions that may not be straightforwardly defined at particular points.
Limits of a Function
Limits of a function are a foundational concept in calculus that allow us to analyze the behavior of functions at specific points or as inputs head towards certain values. A limit captures what happens to \( g(x) \) as \( x \) approaches some value \( a \), even if \( x \) doesn't exactly equal \( a \).

There are several key aspects of limits:

  • Continuity: If the limit of \( g(x) \) as \( x \) approaches \( a \) is exactly \( g(a) \), then the function is said to be continuous at \( a \). This means there are no jumps, gaps, or breaks in the graph of the function at that point.
  • Existence of a Limit: A limit exists if we can make \( g(x) \) arbitrarily close to \( k \) by choosing \( x \) sufficiently close to \( a \), regardless of the direction from which \( x \) approaches \( a \).
  • Left-Hand and Right-Hand Limits: Sometimes the behavior of \( g(x) \) as \( x \) approaches \( a \) from the left (denoted by \( \lim_{x \to a^-} g(x) \)) can be different from the right (denoted by \( \lim_{x \to a^+} g(x) \)). For the overall limit to exist, both of these one-sided limits must be equal.
Limits are essential in determining the properties of functions and provide a deeper understanding of their overall nature and behavior.

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Most popular questions from this chapter

Use a graphing calculator or computer grapher to solve the equations in Exercises \(63-70 .\) \(x^{3}-15 x+1=0 \quad\) (three roots)

Here is the definition of infinite right-hand limit. We say that \(f(x)\) approaches infinity as \(x\) approaches \(x_{0}\) from the right, and write $$\lim _{x \rightarrow x_{0}^{+}} f(x)=\infty$$ if, for every positive real number \(B\) , there exists a corresponding number \(\delta>0\) such that for all \(x\) $$x_{0}B$$ Modify the definition to cover the following cases. a. \(\lim _{x \rightarrow x_{0}-} f(x)=\infty\) b. \(\lim _{x \rightarrow x_{0}^{+}} f(x)=-\infty\) c. \(\lim _{x \rightarrow x_{0}-} f(x)=-\infty\)

In Exercises \(43-46\) , find a function that satisfies the given conditions and sketch its graph. (The answers here are are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.) $$ \lim _{x \rightarrow \pm \infty} f(x)=0, \lim _{x \rightarrow 2^{-}} f(x)=\infty, \text { and } \lim _{x \rightarrow 2^{+}} f(x)=\infty $$

In Exercises \(11-18,\) find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there. $$ h(t)=t^{3}, \quad(2,8) $$

A function discontinuous at every point a. Use the fact that every nonempty interval of real numbers contains both rational and irrational numbers to show that the function $$ f(x)=\left\\{\begin{array}{ll}{1,} & {\text { if } x \text { is rational }} \\\ {0,} & {\text { if } x \text { is rrational }} \\ {0,} & {\text { if } x \text { is irrational }}\end{array}\right. $$ is discontinuous at every point. b. Is \(f\) right-continuous or left-continuous at any point?
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