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Chapter 3: Problem 27

Find the limits. $$\lim _{x \rightarrow+\infty}(1-3 / x)^{x}$$

Short Answer

Expert verified
The limit is \(e^{-3}\).

Step by step solution

01

Identify the Limit Condition

We need to evaluate the limit of \((1 - \frac{3}{x})^x\) as \(x\) approaches infinity. This involves finding how the expression behaves as \(x\) becomes very large.
02

Recognize the Exponential Limit Form

The expression \((1 - \frac{3}{x})^x\) resembles the form \((1 + \frac{u}{n})^n\) which converges to \(e^u\) as \(n\rightarrow \infty\). Here, \(u = -3\) and \(n = x\).
03

Apply the Exponential Limit Formula

Using the known result for the exponential form, \((1 + \frac{u}{n})^n\rightarrow e^u\) as \(n \rightarrow \infty\), we substitute \(u = -3\): \((1 - \frac{3}{x})^x \rightarrow e^{-3}\).
04

Conclude with the Final Limit

Thus, the limit as \(x\) approaches infinity is \(e^{-3}\).

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Key Concepts

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

Exponential Limit Form
In calculus, the exponential limit form is a powerful tool for evaluating limits involving expressions of the type \((1 + \frac{u}{n})^n\) as \(n\) approaches infinity. This particular form is significant because it connects to the mathematical constant \(e\), which represents the base of the natural logarithm.

When a function takes the shape \((1 + \frac{u}{n})^n\), it is known to converge to \(e^u\).
  • The base expression involves a fraction \(\frac{u}{n}\) that becomes smaller as \(n\) grows larger.
  • The expression is raised to the power of \(n\), which also grows towards infinity.
Recognizing this pattern in a limit problem makes it possible to substitute \(e^u\) as the limit result, significantly simplifying calculations. Understanding this shortcut allows for rapid problem-solving and insight into the behavior of complex expressions.
Infinity in Calculus
Infinity is a concept rather than a number in calculus, representing an unbounded quantity. When we talk about limits approaching infinity, we're exploring the behavior of functions as they grow beyond all finite bounds.

Infinity plays a crucial role in calculus, particularly in limit problems, integrals, and series.
  • When \(x\) approaches positive infinity, \(x\) becomes larger without bound.
  • In the limit \(\lim _{x \rightarrow+\infty}(1-3 / x)^{x}\), the \(x\) in the denominator drives the term\(-\frac{3}{x}\) towards zero.
This reduction to near-zero affects the base \((1 - \frac{3}{x})\), modifying it to a form closely resembling \(1^\text{large power}\), where clever calculus techniques like exponential limit forms become essential to find meaningful results.
Convergence to e
When dealing with exponential expressions in limits, convergence to \(e\) is a fundamental concept. This convergence arises naturally when we have expressions of the form \((1 + \frac{u}{n})^n\) and similar patterns.

The process of convergence to \(e\) embodies the principle where compounding with smaller increments, repeated many times, results in growth described by \(e\).
  • Each increase in \(n\) closer to infinity makes the fraction smaller and the exponent larger.
  • The result illustrates how minor gains compound to approximate a value linked with \(e\).
In the exercise we examined, this underlying concept was leveraged to show that \((1 - \frac{3}{x})^x\) converges to \(e^{-3}\) as \(x\) approaches infinity, exemplifying how minor losses compound similarly to minor growths in various contexts.

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Most popular questions from this chapter

Find the limits. $$\lim _{x \rightarrow+\infty} x e^{-x}$$

Find the limits. $$\lim _{x \rightarrow 0^{+}} \tan x \ln x$$

Use the differential \(d y\) to approximate \(\Delta y\) when \(x\) changes as indicated. $$y=x \sqrt{8 x+1} ; \text { from } x=3 \text { to } x=3.05$$

Find the limits. $$\lim _{x \rightarrow 0^{+}} \frac{\ln (\sin x)}{\ln (\tan x)}$$

(a) Show that \(\lim _{x \rightarrow \pi / 2}(\pi / 2-x) \tan x=1\) (b) Show that $$\lim _{x \rightarrow \pi / 2}\left(\frac{1}{\pi / 2-x}-\tan x\right)=0$$ (c) It follows from part (b) that the approximation $$\tan x \approx \frac{1}{\pi / 2-x}$$ should be good for values of \(x\) near \(\pi / 2 .\) Use a calculator to find tan \(x\) and \(1 /(\pi / 2-x)\) for \(x=1.57 ;\) compare the results.
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