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Chapter 8: Problem 12

Show that \(\lim \int_{1}^{2} e^{-n x^{2}} d x=0\).

Short Answer

Expert verified
The \(\lim \int_{1}^{2} e^{-n x^{2}} d x=0\). Because as n approaches infinity, \(e^{-n x^{2}}\) tends towards zero, making the area under the curve (and thus the integral) become zero as well.

Step by step solution

01

Evaluate the Integral

To find the integral of a given function such as \(\int_{1}^{2} e^{-n x^{2}} d x\), we must first recognize that the antiderivative of \(e^{-n x^{2}}\) is not directly calculable, instead we can use the properties of definite integrals, so we look at the bounds of the integral. The integral from 1 to 2 of \(e^{-n x^{2}}\) is equal to the area under the \(e^{-n x^{2}}\) curve between x=1 and x=2. As n increases towards infinity, this area becomes increasingly small as the graphs of \(e^{-n x^{2}}\) tend towards a flat line on x-axis, reducing the area under the curve nearly zero.
02

Find the Limit

The limit as n approaches infinity of a function such as \(e^{-n x^{2}}\) gives us an indication of how the function behaves as n becomes very large, or in other words, how it behaves at its bounds. As n approaches infinity in this case, the function \(e^{-n x^{2}}\) approaches zero because \(e^{negative\, large\, number}\) is essentially zero. Thus, we can say that the limit as n tends towards infinity of the integral from 1 to 2 of \(e^{-n x^{2}}\) is essentially zero. This is because the integral represents area under the curve and as n tends towards infinity, this area decreases to 0 resulting in the limit being 0.

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Key Concepts

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

Definite Integrals
Definite integrals are a fundamental concept in calculus. They help us calculate the exact area under a curve from one point to another. When we see an expression like \[ \int_{a}^{b} f(x) \, dx \], it signifies the entire region under the curve of \( f(x) \) between the limits \( x = a \) and \( x = b \).

  • The function \( f(x) \) forms a curve above the x-axis.
  • The limits \( a \) and \( b \) are the bounds of integration.
  • The result of the integral is the net area between the curve and the x-axis, considering areas below the x-axis as negative.
Now, when we deal with an integral such as \( \int_{1}^{2} e^{-n x^{2}} \, dx \), it represents this process for the specific curve of \( e^{-n x^{2}} \) from \( x = 1 \) to \( x = 2 \). As the parameter \( n \) increases, the curve becomes increasingly steep and flat around the x-axis, which tends to shrink the area under the curve, eventually reducing it to nearly zero.
Limit of a Function
The concept of limits is crucial when dealing with functions that involve variables approaching a particular value. Limits help us understand the behavior of functions as they near a certain point.

In this context, examining the limit \( \lim_{n \to \infty} \int_{1}^{2} e^{-n x^{2}} \, dx \) means observing how the area under the curve changes as \( n \) increases indefinitely.
  • A limit helps describe what happens to a function's value as the input approaches a particular point or infinity.
  • In our exercise, as \( n \) becomes very large, the function \( e^{-n x^{2}} \) becomes extremely close to zero everywhere between \( x = 1 \) and \( x = 2 \).
  • This behavior allows us to conclude that the integral, which represents the area, approaches zero.
Thus, limits provide a way to calculate outcomes when direct evaluation is impossible, offering insights into the tendencies of functions as parameters change.
Exponential Functions
Exponential functions are characterized by a constant base raised to a variable exponent, like \( e^{-n x^{2}} \). These functions are known for their rapid growth or decay, depending on the sign of the exponent. In our exercise, the expression involves \(-n x^{2}\) in the exponent leading to a decay.

  • Exponential functions with a negative exponent, like \( e^{-n x^{2}} \), decrease swiftly as \( x \) increases.
  • The function \( e^{negative\, number} \) yields a value close to zero.
  • As \( n \) increases, the decay of \( e^{-n x^{2}} \) becomes more prominent, making the function's impact on the integral diminish.
This steep decline in \( e^{-n x^{2}} \) affects the definite integral by moving the graph progressively closer to the x-axis, shrinking the area under the curve between any two points to zero as \( n \) reaches infinity.

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Most popular questions from this chapter

Let \(g_{n}(x):=e^{-n x} / n\) for \(x \geq 0, n \in \mathbb{N}\). Examine the relation between \(\lim \left(g_{n}\right)\) and \(\lim \left(g_{n}^{\prime}\right)\).

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