/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 Let \(g_{n}(x):=e^{-n x} / n\) f... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 10

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 \(\lim \left(g_{n}^{\prime}\right)\).

Short Answer

Expert verified
Both \(\lim_{n \rightarrow \infty} g_{n}(x) = 0\) and \(\lim_{n \rightarrow \infty} g_{n}^{\prime}(x) = 0\), hence showing that the limit of the function and its derivative are both equal to zero as \(n \rightarrow \infty\).

Step by step solution

01

Understand the Function

First, let's understand the function \(g_{n}(x) = \frac {e^{-n x}}{n}\). Here \(n\) is a natural number, and \(x\) is a real number, for which the function is defined, greater than or equal to zero.
02

Calculate the Limit of the Function

We need to find \(\lim_{n \rightarrow \infty} g_{n}(x)\). For each fixed \(x \geq 0\), we have \(\lim_{n \rightarrow \infty} \frac {e^{-n x}}{n} = 0\) (because \(e^{-n x} \rightarrow 0\) much faster than \(n \rightarrow \infty\). Therefore, \(\lim_{n \rightarrow \infty} g_{n}(x) = 0\).
03

Find the Derivative of the Function

Next, let's find the derivative \(g_{n}^{\prime}(x)\) of the function. By using the chain rule, we obtain \(g_{n}^{\prime}(x) = \frac {-e^{-n x}}{n}\).
04

Calculate the Limit of the Derivative Function

We then find \(\lim_{n \rightarrow \infty} g_{n}^{\prime}(x)\). Similar to Step 2, for each fixed \(x \geq 0\), we have \( \lim_{n \rightarrow \infty} \frac {-e^{-n x}}{n} = 0\). Therefore, \(\lim_{n \rightarrow \infty} g_{n}^{\prime}(x) = 0\).

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Most popular questions from this chapter

Evaluate \(\lim \left(x^{n} /\left(1+x^{n}\right)\right)\) for \(x \in \mathbb{R}, x \geq 0\)

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