Problem 45
Question
The value of \(\lim _{n \rightarrow \infty}\left[\sqrt[3]{n^{2}-n^{3}}+n\right]\) is (A) \(\frac{1}{3}\) (B) \(-\frac{1}{3}\) (C) \(\frac{2}{3}\) (D) \(-\frac{2}{3}\)
Step-by-Step Solution
Verified Answer
The limit is \(-\frac{1}{3}\).
1Step 1: Simplify the expression under the cube root
Consider the limit expression \( \sqrt[3]{n^2 - n^3} + n \). Simplify the expression under the cube root: \( n^2 - n^3 = n^2(1 - n) = -n^3(1 - \frac{1}{n}) \).
2Step 2: Rewrite the cube root
Now express \( \sqrt[3]{-n^3(1 - \frac{1}{n})} \). Split this into \( \sqrt[3]{-n^3} \cdot \sqrt[3]{1 - \frac{1}{n}} \). The cube root of \( -n^3 \) is \( -n \).
3Step 3: Apply the limit to each part
Now consider the expression as \( -n \cdot \sqrt[3]{1 - \frac{1}{n}} + n \). Simplify the expression: \( -n + n \cdot ( 1 - \frac{1}{3n}) \).
4Step 4: Evaluate the expression's limit
Apply the limit as \( n \to \infty \): The \( n \) terms cancel as \( n - n = 0\), and the remaining expression becomes \( n \cdot \left(0 - \frac{1}{3n}\right) = -\frac{1}{3} \).
5Step 5: Conclude with the correct limit
Hence, the value of the limit is calculated as \(-\frac{1}{3}\).
Key Concepts
Asymptotic AnalysisInfinite LimitsSimplification of Expressions
Asymptotic Analysis
Asymptotic analysis is a crucial concept in calculus and mathematics that deals with understanding the behavior of functions as variables approach certain limits, often infinity. It is like peeking into the distant future of a function’s behavior.
In this sense, asymptotic analysis helps to simplify complex expressions and evaluate limits by focusing on the most significant terms as variables grow large.
For example, in the given exercise, the expression under the cube root becomes extremely large as \( n \) becomes infinite. Hence, to understand the core behavior, we extract the leading term \(-n^3\) because it dominates other terms and dictates the behavior of the expression as \( n \to \infty \).
In this sense, asymptotic analysis helps to simplify complex expressions and evaluate limits by focusing on the most significant terms as variables grow large.
For example, in the given exercise, the expression under the cube root becomes extremely large as \( n \) becomes infinite. Hence, to understand the core behavior, we extract the leading term \(-n^3\) because it dominates other terms and dictates the behavior of the expression as \( n \to \infty \).
- By focusing on \(-n^3\), asymptotic analysis streamlines the problem-solving process, allowing us to simplify and predict the function's behavior in less complex terms.
- It’s particularly beneficial in solving limits or finding growth rates, making complicated functions more manageable.
Infinite Limits
Infinite limits describe the behavior of a function or an expression as its input approaches infinity. Instead of converging to a finite value, the function may infinitely increase or decrease, or oscillate.
In our exercise, the expression \( \lim _{n \rightarrow \infty}\left[\sqrt[3]{n^{2}-n^{3}}+n\right] \) represents a limit as \( n \) heads towards infinity.
To understand an infinite limit, it’s essential to consider the dominant behavior as variables grow vastly large.
Therefore, understanding infinite limits facilitates evaluating limits of complex expressions and their behavior as they diverge into infinity areas.
In our exercise, the expression \( \lim _{n \rightarrow \infty}\left[\sqrt[3]{n^{2}-n^{3}}+n\right] \) represents a limit as \( n \) heads towards infinity.
To understand an infinite limit, it’s essential to consider the dominant behavior as variables grow vastly large.
- Rewriting the cube root using leading expressions helps simplify the calculation of \( \sqrt[3]{-n^3(1 - \frac{1}{n})} \) to \(-n\).
- This makes the analysis straightforward, focusing on the cancelling nature of \(-n\) and \(n\) terms and the remaining small fraction \( \left(0 - \frac{1}{3n}\right) \).
Therefore, understanding infinite limits facilitates evaluating limits of complex expressions and their behavior as they diverge into infinity areas.
Simplification of Expressions
Simplification of expressions is a technique used in calculus to reduce complexity by breaking down equations or expressions into more manageable forms.
In the original exercise, simplification helps in unraveling \( \sqrt[3]{n^{2}-n^{3}}+n \) into separate, approachable components. By focusing on the dominant terms, intuitive simplification methods guide the solution towards evaluating limits more effectively.
In the original exercise, simplification helps in unraveling \( \sqrt[3]{n^{2}-n^{3}}+n \) into separate, approachable components. By focusing on the dominant terms, intuitive simplification methods guide the solution towards evaluating limits more effectively.
- For instance, expressing the cube root as \( -n \cdot \sqrt[3]{1 - \frac{1}{n}} \) simplifies and highlights the primary behaviour of the function.
- Moreover, once you resolve that \( -n + n \cdot (1 - \frac{1}{3n}) \) simplifies down to \( -\frac{1}{3} \) as other terms cancel out, determination becomes more straightforward.
Other exercises in this chapter
Problem 43
\(\lim _{x \rightarrow \infty} \sqrt{\frac{x+\sin x}{x-\cos x}}=\) (A) 0 (B) 1 (C) \(-1\) (D) None of these
View solution Problem 44
If \(S_{n}=\sum_{i=1}^{n} a_{i}\) and \(\lim _{n \rightarrow \infty} a_{n}=a\), then \(\lim _{n \rightarrow \infty} \frac{S_{n+1}-S_{n}}{\sqrt{\sum_{i=1}^{n} i}
View solution Problem 46
The value of \(\lim _{n \rightarrow \infty} \frac{\sqrt[4]{n^{5}+2}-\sqrt[3]{n^{2}+1}}{\sqrt[5]{n^{4}+2}-\sqrt[2]{n^{3}+1}}\) is (A) 1 (B) 0 (C) \(-1\) (D) \(\i
View solution Problem 47
The integer \(n\) for which \(\lim _{x \rightarrow 0} \frac{(\cos x-1)\left(\cos x-e^{x}\right)}{x^{n}}\) is a finite non-zero number, is (A) 1 (B) 2 (C) 3 (D)
View solution