/*! 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 33 Find the limits. $$ \lim _{x... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 33

Find the limits. $$ \lim _{x \rightarrow-\infty} \frac{1-e^{x}}{1+e^{x}} $$

Short Answer

Expert verified
The limit is 1.

Step by step solution

01

Substitute a Limit Approach Direction

To solve the limit \( \lim_{x \to -\infty} \frac{1 - e^x}{1 + e^x} \), we need to understand the behavior of the expression as \( x \) approaches \( -\infty \). This means substituting values that make \( x \) very negative and observe the behavior.
02

Analyze Exponential Behavior

As \( x \to -\infty \), the exponential function \( e^x \) approaches 0 because \( e^x \) grows rapidly towards zero for very negative values of \( x \). Therefore, \( e^x \approx 0 \).
03

Simplify the Expression

Replace \( e^x \) with 0 in the limit expression to approximate the behavior:\[\frac{1 - e^x}{1 + e^x} \approx \frac{1 - 0}{1 + 0} = \frac{1}{1}\]This simplifies to 1.
04

Conclude the Limit

Since as \( x \to -\infty \), \( e^x \to 0 \), the limit simplifies directly:\[\lim_{x \to -\infty} \frac{1 - e^x}{1 + e^x} = 1\]

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Key Concepts

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

Exponential Function
An exponential function is a mathematical function of the form \( f(x) = a \, e^{bx} \), where \( e \) is approximately equal to 2.71828, and it is known as Euler's number. The values \( a \) and \( b \) are constants that control the function's base and growth rate, respectively. A key characteristic of exponential functions is their rapid growth.
  • If \( b > 0 \), the function grows exponentially as \( x \) increases, meaning it gets larger very quickly.
  • If \( b < 0 \), as in our problem where \( e^x \) is considered with \( x \to -\infty \), the function decreases rapidly approaching zero.
Exponential functions appear in various real-world applications like population growth, radioactive decay, and interest calculations. They are quite unique due to their consistent relative growth rate, meaning they increase by a constant percentage over equal incremented intervals.
Limits at Infinity
When considering a function's behavior as \( x \) approaches infinity (or negative infinity), we are analyzing its limit at infinity. It involves observing how the function behaves for extremely large positive or negative values of \( x \).
  • The limit informs us about the end behavior of a function, describing what the values of the function approach as \( x \) heads towards infinity or negative infinity.
  • Mathematically, for the expression \( \lim_{x \rightarrow -\infty} \frac{1 - e^{x}}{1 + e^{x}} \), the challenge is to find out what happens to the entire expression as \( x \) keeps decreasing.
In our example, \( e^x \to 0 \) as \( x \to -\infty \), causing the whole fraction to simplify greatly, reaching the value of 1. This simplification occurs because both the numerator and denominator trend towards constant values, essentially "canceling out" the influence of \( e^x \) in the expression.
Behavior of Exponential Functions
The behavior of exponential functions is generally characterized by their base \( e \) and the exponent鈥檚 influence. As noted, exponential functions like \( e^x \) increase or decrease based on the sign of the exponent. Here are key aspects:
  • Positive Exponents: They result in a rapid increase, where the function grows significantly larger as \( x \) becomes larger.
  • Negative Exponents: They lead to a sharp decrease, causing the function to approach zero, as observed in \( e^x \) when \( x \to -\infty \).
Studying their behavior helps to predict and understand long-term trends and tendencies of these functions. For instance, knowing \( e^x \to 0 \) as \( x \to -\infty \) allows us to readily simplify limits involving exponential terms, determining outcomes such as in our problem where the expression stabilizes at a limit of 1.

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Most popular questions from this chapter

Evaluate the limit using an appropriate substitution. $$ \lim _{x \rightarrow+\infty}\left(1+\frac{2}{x}\right)^{x}[\text { Hint }: t=x / 2] $$

Evaluate the limit using an appropriate substitution. $$ \lim _{x \rightarrow 0^{-}} e^{\csc x} $$

In each part, find the largest open interval centered at \(x=1\), such that for each \(x\) in the interval, other than the center, the value of \(f(x)=1 /(x-1)^{2}\) is greater than \(M\). (a) \(M=10\) (b) \(M=1000\) (c) \(M=100,000\)

True-False Determine whether the statement is true or false. Explain your answer. Suppose that for all real numbers \(x\), a function \(f\) satisfies $$ |f(x)+5| \leq|x+1| $$ Then \(\lim _{x \rightarrow-1} f(x)=-5\).

Evaluate the limit using an appropriate substitution. $$ \lim _{x \rightarrow 0^{-}} e^{1 / x} $$
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