Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 8: Problem 26
Find the limit of the following sequences or determine that the limit does not exist. $$\left\\{\left(1-\frac{4}{n}\right)^{n}\right\\}$$
Short Answer
Expert verified
Answer: The limit of the sequence is \(e^{\frac{1}{4}}\).
Step by step solution
01
Write the sequence as a limit of a function
To solve this problem, let's consider the function \(f(x)=\left(1-\frac{4}{x}\right)^x\). Then we can express the sequence as the limit of the function as \(x\) approaches infinity:
$$\lim_{n \to \infty} \left\{\left(1-\frac{4}{n}\right)^{n}\right\} = \lim_{x \to \infty} \left(1-\frac{4}{x}\right)^x$$
02
Apply the limit properties for the exponential functions
We can rewrite the limit using the natural exponential function by using the fact that \(e^{\ln(a)}=a\) for all \(a>0\). Thus, we have:
$$\lim_{x \to \infty} \left(1-\frac{4}{x}\right)^x = \lim_{x \to \infty} e^{\ln\left(\left(1-\frac{4}{x}\right)^x\right)}$$
Now, apply the logarithm property \(\ln(a^b)=b\ln(a)\), we get:
$$\lim_{x \to \infty} e^{\ln\left(\left(1-\frac{4}{x}\right)^x\right)} = \lim_{x \to \infty} e^{x\cdot\ln\left(1-\frac{4}{x}\right)}$$
03
Evaluate the limit inside the exponential
Now focus on the limit inside the exponential function:
$$\lim_{x \to \infty} x\cdot\ln\left(1-\frac{4}{x}\right)$$
Apply the well-known trick with the limit \(x\cdot\ln(1+c)\) and substitute \(t=\frac{4}{x}\) to rewrite the limit as follows:
$$\lim_{x \to \infty} x\cdot\ln\left(1-\frac{4}{x}\right) = \lim_{t\to 0}\frac{\ln(1-t)}{-4t}$$
04
Apply L'Hopital's Rule to this limit
Now we can use L'Hopital's Rule to evaluate the limit since it is in the indeterminate form \(\frac{0}{0}\):
$$\lim_{t \to 0}\frac{\ln(1-t)}{-4t} =\lim_{t \to 0} \frac{\frac{d}{dt}[\ln(1-t)]}{\frac{d}{dt}[-4t]}$$
Taking the derivatives with respect to \(t\), we get:
$$\lim_{t \to 0}\frac{\frac{-1}{1-t}}{-4} =\lim_{t \to 0}\frac{1}{4(1-t)}=\frac{1}{4}$$
05
Compute the limit of the entire sequence
Now substitute the result from step 4 back into our limit of the exponential function and compute the final answer:
$$\lim_{n \to \infty} \left\{\left(1-\frac{4}{n}\right)^{n}\right\} = \lim_{x \to \infty} e^{x\cdot\ln\left(1-\frac{4}{x}\right)}=e^{\frac{1}{4}}$$
So the limit of the sequence is \(\boxed{e^{\frac{1}{4}}}\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Functions
Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. They are defined as functions of the form \( f(x) = a^x \), where \( a \) is the base and \( x \) is the exponent.
In this context, when dealing with limits of sequences, it becomes essential to express the sequence in terms of exponential functions to simplify evaluation. Transforming the original sequence \( \left(1-\frac{4}{x}\right)^x \) into \( e^{\ln\left(\left(1-\frac{4}{x}\right)^x\right)} \) allows us to leverage properties of both exponential functions and logarithms.
This transformation highlights the rapid growth or decay of terms in the sequence as \( x \) approaches infinity, making it much easier to identify the limit behavior. Exponential functions thus play a crucial role in simplifying and solving complex limit problems, revealing a deeper understanding of the asymptotic behavior of sequences.
In this context, when dealing with limits of sequences, it becomes essential to express the sequence in terms of exponential functions to simplify evaluation. Transforming the original sequence \( \left(1-\frac{4}{x}\right)^x \) into \( e^{\ln\left(\left(1-\frac{4}{x}\right)^x\right)} \) allows us to leverage properties of both exponential functions and logarithms.
This transformation highlights the rapid growth or decay of terms in the sequence as \( x \) approaches infinity, making it much easier to identify the limit behavior. Exponential functions thus play a crucial role in simplifying and solving complex limit problems, revealing a deeper understanding of the asymptotic behavior of sequences.
L'Hopital's Rule
L'Hopital's Rule is a valuable tool for solving limits involving indeterminate forms such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). It states that if you have a limit of the form \( \lim_{x \to c} \frac{f(x)}{g(x)} \), where both \( f(x) \) and \( g(x) \) approach zero or infinity as \( x \) approaches \( c \), this limit can be evaluated by differentiating the numerator and the denominator:
\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \]
provided the limit of the derivatives exists.
In our sequence limit problem, after substituting \( t = \frac{4}{x} \) and rewriting \( x \cdot \ln\left(1-\frac{4}{x}\right) \) as \( \frac{\ln(1-t)}{-4t} \), we encounter an indeterminate form at \( t \to 0 \). By applying L'Hopital's Rule, we are able to differentiate the numerator \( \ln(1-t) \) and denominator \( -4t \) to obtain a simpler expression, yielding \( \frac{1}{4} \) as the evaluated limit. This result lets us successfully handle the indeterminacy and find the exponential limit of the original sequence.
\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \]
provided the limit of the derivatives exists.
In our sequence limit problem, after substituting \( t = \frac{4}{x} \) and rewriting \( x \cdot \ln\left(1-\frac{4}{x}\right) \) as \( \frac{\ln(1-t)}{-4t} \), we encounter an indeterminate form at \( t \to 0 \). By applying L'Hopital's Rule, we are able to differentiate the numerator \( \ln(1-t) \) and denominator \( -4t \) to obtain a simpler expression, yielding \( \frac{1}{4} \) as the evaluated limit. This result lets us successfully handle the indeterminacy and find the exponential limit of the original sequence.
Natural Logarithm
The natural logarithm, denoted as \( \ln(x) \), is a logarithm with the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. The natural logarithm is particularly useful in mathematics because it simplifies derivatives and integrals involving exponential functions.
In solving sequence limit problems, using \( \ln(x) \) can help rewrite exponential expressions in a form that is easier to analyze. For our sequence, we used the property \( a^b = e^{b \cdot \ln(a)} \) to express \( \left(1-\frac{4}{x}\right)^x \) as \( e^{x \cdot \ln\left(1-\frac{4}{x}\right)} \).
By taking the natural logarithm, we linearize the exponentiation, converting multiplication and exponentiation into addition and multiplication. This transformation not only simplifies limit processes but also unveils new perspectives on the sequence's behavior, driving a deeper understanding of the relationship between exponential and logarithmic functions. Analyzing these properties allows students to tackle more complex problems with confidence and ease.
In solving sequence limit problems, using \( \ln(x) \) can help rewrite exponential expressions in a form that is easier to analyze. For our sequence, we used the property \( a^b = e^{b \cdot \ln(a)} \) to express \( \left(1-\frac{4}{x}\right)^x \) as \( e^{x \cdot \ln\left(1-\frac{4}{x}\right)} \).
By taking the natural logarithm, we linearize the exponentiation, converting multiplication and exponentiation into addition and multiplication. This transformation not only simplifies limit processes but also unveils new perspectives on the sequence's behavior, driving a deeper understanding of the relationship between exponential and logarithmic functions. Analyzing these properties allows students to tackle more complex problems with confidence and ease.
