Problem 95 - Single Variable Calculus

Question

Suppose that \(f(x)

Step-by-Step Solution

Answer

Yes, it follows that \(\lim _{x \rightarrow a} f(x)<\lim _{x \rightarrow a} g(x)\) under the given conditions. This is because we established that the inequality \(f(x) < g(x)\) for all x in an open interval containing \(a\) (except possibly at a), can be used alongside the existence of the limits of both functions to show that \(L

Step 1: Statement of the Problem

We are given two functions, f(x) and g(x), such that for all x in an open interval containing a except possibly at a, f(x) < g(x). We are also given that both \(\lim _{x \rightarrow a} f(x)\) and \(\lim _{x \rightarrow a} g(x)\) exist. We need to determine if it follows that \(\lim _{x \rightarrow a} f(x)<\lim _{x \rightarrow a} g(x)\).

Step 2: Analyze the Given Conditions

We have the following conditions: 1. f(x) < g(x) for all x in an open interval containing a, except possibly at a. 2. \(\lim _{x \rightarrow a} f(x)\) exists. 3. \(\lim _{x \rightarrow a} g(x)\) exists. Now we will proceed to show the desired inequality between the limits based on these given conditions.

Step 3: Showing the Inequality between Limits

Let L and M be the respective limits of f(x) and g(x) as x approaches a. That is, \[ L = \lim _{x \rightarrow a} f(x) \] \[ M = \lim _{x \rightarrow a} g(x) \] Now, since the limits exist, given any \(\epsilon > 0\), there exist \(\delta_1 > 0\) and \(\delta_2 > 0\) such that: 1. If \( 0 < |x - a| < \delta_1\), then \(|f(x) - L| < \frac{\epsilon}{2}\). 2. If \( 0 < |x - a| < \delta_2\), then \(|g(x) - M| < \frac{\epsilon}{2}\). Let \(\delta = \min(\delta_1, \delta_2)\). Consider any x such that \(0 < |x - a| < \delta\). Then from the given conditions, we have \[ f(x) < g(x) \] Now by definition of delta, we have \[ |f(x) - L| < \frac{\epsilon}{2} \] \[ |g(x) - M| < \frac{\epsilon}{2} \] Now we aim to show that \(L < M \). Note that, \[ g(x) - f(x) > 0 \] \[ g(x) - M + M - f(x) > 0 \] \[ (g(x) - M) + (M - f(x)) > 0 \] By triangle inequality, \[ |g(x) - M| + |M - f(x)| > 0 \] We have, \[ |g(x) - M| + |L - f(x)| = |g(x) - M| + |f(x) - L| \] Now since, \[ |f(x) - L| < \frac{\epsilon}{2} \] \[ |g(x) - M| < \frac{\epsilon}{2}\] We get, \[ L - M < \frac{\epsilon}{2} + \frac{\epsilon}{2} \] \[ L - M < \epsilon \] Since \(\epsilon\) is arbitrary, this implies \(L \leq M\). But we know that for every x in the given interval, f(x) < g(x). So, we deduce that \(L < M\), and we get \[ \lim _{x \rightarrow a} f(x)<\lim _{x \rightarrow a} g(x) \] This concludes the proof.

Key Concepts

Limits

In calculus, limits help us understand the behavior of a function as it approaches a particular point. A limit essentially tells us what value the output of the function gets closer to as the input approaches a certain point.
When we talk about the limit of a function \(f(x)\) as \(x\) approaches \(a\), we are interested in finding the value that \(f(x)\) is getting closer to — even if \(x\) never actually reaches \(a\). The notation used to denote this is \( \lim_{x \to a} f(x) \). This concept is crucial when the function isn't defined at \(x = a\) or behaves differently around that point.
There are certain conditions necessary for a limit to exist. Primarily, as \(x\) approaches \(a\), the function values must approach a specific number from both sides. This is important in understanding the problem, where we study two functions approaching different limits.

Inequalities

Inequalities are expressions that show the relationship of one quantity being greater or smaller than another. In the context of functions and limits, they allow us to compare the behavior of two functions as their inputs approach a particular value.
In this exercise, you are provided that \(f(x) < g(x)\) for all \(x\) in an open interval around \(a\), except possibly at \(a\). This inequality suggests that \(f(x)\) consistently has smaller values compared to \(g(x)\). Understanding inequalities helps in comparing the endpoint behavior of functions.
When dealing with limits, even if \(f(x)\) is less than \(g(x)\) throughout an interval, we need to prove if the same applies to their limits. The problem utilizes these principles to demonstrate that if such an inequality is present, then the inequality \(\lim_{x \to a} f(x) < \lim_{x \to a} g(x)\) holds true.

Functions

Functions form the backbone of calculus and mathematics as a whole. Essentially, a function defines a particular relationship between a set of inputs and a set of permissible outputs. In this exercise, \(f(x)\) and \(g(x)\) are the two functions under comparison.
Understanding the behavior of functions is crucial in calculus, especially when examining how they behave near certain points or over particular intervals. Functions can be represented graphically, visually showing how values change.
The exercise investigates the behavior of two different functions in terms of their limits. Note that analyzing functions often involves understanding certain terms or notations used in calculus, like the limit \(\lim_{x \rightarrow a} f(x)\). This gives valuable insight into how functions behave as the input nears a particular value, helping to resolve queries about inequalities of their limits.

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Problem 95

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Problem 98

Other exercises in this chapter

Problem 94

Show by means of an example that \(\lim _{x \rightarrow a a}[f(x) g(x)]\) may exist even though neither \(\lim _{x \rightarrow a} f(x)\) nor \(\lim _{x \rightar

View solution

Problem 95

Prove that if \(f\) and \(g\) are continuous at \(a\) with \(g(a) \neq 0\), then \(f / g\) is continuous at \(a\).

View solution

Problem 98

Show that \(\lim _{x \rightarrow a} \cos x=\cos a\). (See the hint for Exercise 97.)

View solution

Problem 99

In Exercises 99-102, determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that shows it

View solution

Key Concepts

Limits

In calculus, limits help us understand the behavior of a function as it approaches a particular point. A limit essentially tells us what value the output of the function gets closer to as the input approaches a certain point.
When we talk about the limit of a function \(f(x)\) as \(x\) approaches \(a\), we are interested in finding the value that \(f(x)\) is getting closer to — even if \(x\) never actually reaches \(a\). The notation used to denote this is \( \lim_{x \to a} f(x) \). This concept is crucial when the function isn't defined at \(x = a\) or behaves differently around that point.
There are certain conditions necessary for a limit to exist. Primarily, as \(x\) approaches \(a\), the function values must approach a specific number from both sides. This is important in understanding the problem, where we study two functions approaching different limits.

Inequalities

Inequalities are expressions that show the relationship of one quantity being greater or smaller than another. In the context of functions and limits, they allow us to compare the behavior of two functions as their inputs approach a particular value.
In this exercise, you are provided that \(f(x) < g(x)\) for all \(x\) in an open interval around \(a\), except possibly at \(a\). This inequality suggests that \(f(x)\) consistently has smaller values compared to \(g(x)\). Understanding inequalities helps in comparing the endpoint behavior of functions.
When dealing with limits, even if \(f(x)\) is less than \(g(x)\) throughout an interval, we need to prove if the same applies to their limits. The problem utilizes these principles to demonstrate that if such an inequality is present, then the inequality \(\lim_{x \to a} f(x) < \lim_{x \to a} g(x)\) holds true.

Functions

Functions form the backbone of calculus and mathematics as a whole. Essentially, a function defines a particular relationship between a set of inputs and a set of permissible outputs. In this exercise, \(f(x)\) and \(g(x)\) are the two functions under comparison.
Understanding the behavior of functions is crucial in calculus, especially when examining how they behave near certain points or over particular intervals. Functions can be represented graphically, visually showing how values change.
The exercise investigates the behavior of two different functions in terms of their limits. Note that analyzing functions often involves understanding certain terms or notations used in calculus, like the limit \(\lim_{x \rightarrow a} f(x)\). This gives valuable insight into how functions behave as the input nears a particular value, helping to resolve queries about inequalities of their limits.