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Chapter 5: Problem 57

The function \(f(x)=\left\\{\begin{array}{ll}{\frac{1}{x^{2}},} & {0

Short Answer

Expert verified
(a) The graph of \(f(x)\) approaches \(\infty\) as \(x\) approaches 0 from the right. (b) The function \(f(x)\) isn't integrable on the interval \([0,1]\) because the Riemann sum does not converge to a finite value, specifically because \(f(x)\) approaches \(\infty\) as \(x\) approaches 0 from the right.

Step by step solution

01

Understanding the Limit

For part (a), \(f(x)\) for \(0<x\leq1\) is \(\dfrac{1}{x^{2}}\). So as \(x\) approaches 0 from the right (i.e. \(x\) tends to 0+), \(f(x)\) tends to \(\infty\). Thus, the graph shoots up to infinity as \(x\) gets close to 0 from the right.
02

Understanding Riemann Sums

For part (b), recall that for a function to be Riemann integrable, the limit of the Riemann sums must exist as the partition of the interval becomes finer. A Riemann sum approximates the area under the curve by summing up the areas of rectangles.
03

Using Riemann Sums for the Function

The height of each rectangle in the Riemann sum is determined by the value of the function at that point. For \(f(x)\), as \(x\) gets close to 0 from the right, the value skyrockets to \(\infty\). So, no matter how fine our partition is, there's going to be at least one rectangle with a height tending to \(\infty\). Hence, the area under the curve cannot be a fixed finite number.
04

Function Non-Integrability Conclusion

Therefore, based on Riemann sums, a convincing argument can be made that the function \(f(x)\) is not integrable on \([0,1]\) because the sum fails to converge to a finite limit (since part of the function tends to \(\infty\)).

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Key Concepts

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

Limit of a Function
The concept of the limit of a function is foundational in calculus. It describes the behavior of a function as the input approaches a certain value. For instance, when exploring the limit of the function \( f(x) = \frac{1}{x^2} \) for \( 0 < x \leq 1 \) as \( x \) approaches 0, one observes that \( f(x) \) becomes arbitrarily large. The notation \( \lim_{x \to 0^+} f(x) = \infty \) represents this observation, indicating an infinite limit. Limits help us understand the behavior of functions near points of interest, including points that might not be within the domain of the function.

Discontinuity in Calculus
A discontinuity is a point at which a function is not continuous. At such points, there is an abrupt change in the value of the function. There are different types of discontinuities, such as jump, infinite, and removable. The function in our exercise, \( f(x) \) has an infinite discontinuity at \( x=0 \), meaning the function value heads towards infinity as \( x \) approaches 0 from the right. Discontinuities are critical to identify because they can affect the behavior of functions and the ability to perform certain calculus operations on them, like integration.

Non-Integrable Functions
Some functions, referred to as non-integrable functions, cannot be integrated over a certain interval using standard analytical techniques. This typically happens when the limit of the Riemann sums does not exist, representing the total area under a curve. The function in our example, \( f(x) \) becomes infinitely large as \( x \) approaches 0, leading to Riemann sums that also tend towards infinity, regardless of partition size. Such functions can pose interesting challenges and require alternative methods or interpretations to deal with their behavior at points of discontinuity.

Infinite Limits
An infinite limit describes a situation where the output of a function grows without bounds as the input approaches a specific value. This is not the same as saying the limit is infinity, as infinity is not a number, but a concept of unboundedness. In our example, as \( x \) approaches 0 from the right, the output \( f(x) = \frac{1}{x^2} \) grows without bounds. Infinite limits often occur at points of discontinuity and have substantial implications in calculus, notably affecting the convergence or divergence of integrals and series.

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