Problem 24
Question
15-36 Find the limit. $$ \lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{6}-x}}{x^{3}+1} $$
Step-by-Step Solution
Verified Answer
The limit is 3.
1Step 1: Analyze the expression
We need to find the limit as \( x \) approaches infinity for \( \frac{\sqrt{9x^{6} - x}}{x^{3} + 1} \). Start by analyzing the expressions inside the numerator and the denominator.
2Step 2: Simplify using dominant terms
Identify the dominant term in the numerator: \( \sqrt{9x^{6} - x} \approx \sqrt{9x^{6}} = 3x^3 \). Similarly, in the denominator the dominant term is \( x^3 \). Simplify the expression considering these terms.
3Step 3: Simplify the entire expression
Replace the numerator and denominator by their dominant terms so the expression becomes \( \frac{3x^3}{x^3} \).
4Step 4: Calculate the limit
The expression reduces to \( 3 \) as both \( x^3 \) terms cancel each other out. Since the simplified expression is a constant, the limit as \( x \rightarrow \infty \) is simply this constant: \( 3 \).
Key Concepts
Dominant TermsAsymptotic AnalysisInfinite Limits
Dominant Terms
Understanding dominant terms is crucial when calculating limits, especially as variables approach infinity. In mathematical functions with multiple terms, the dominant term has the greatest influence on the function's behavior as variables grow larger. This dominance is often visible as the term with the highest degree in a polynomial or the term with a variable raised to the largest exponent.
Imagine a polynomial function like \( ax^n + bx^{n-1} + \, ... \, + cx + d \). Here, if \( x \) gets very large, the term \( ax^n \) will have the most significant impact on the value of the whole polynomial. Thus, we say \( ax^n \) is the dominant term. Concretely, this term "dominates" because it grows much faster than the others, overshadowing them as \( x \) increases.
Imagine a polynomial function like \( ax^n + bx^{n-1} + \, ... \, + cx + d \). Here, if \( x \) gets very large, the term \( ax^n \) will have the most significant impact on the value of the whole polynomial. Thus, we say \( ax^n \) is the dominant term. Concretely, this term "dominates" because it grows much faster than the others, overshadowing them as \( x \) increases.
- Dominant terms simplify complex expressions.
- They help predict the behavior of expressions for large values of variables.
- Simplifying by dominant terms makes it easier to compute limits.
Asymptotic Analysis
Asymptotic analysis is a tool used to describe the behavior of functions as they tend toward a specified limit, often infinity. In calculus, it aids in understanding how functions behave as their input values become very large or very small, far from their starting point.
To conduct an asymptotic analysis, we primarily focus on the rate at which the functions grow or decay. This approach involves comparing functions and determining which terms will affect the function most significantly in the limit. Asymptotically, some terms in a function might become negligible when others become infinitely large or small, which allows for simplification.
To conduct an asymptotic analysis, we primarily focus on the rate at which the functions grow or decay. This approach involves comparing functions and determining which terms will affect the function most significantly in the limit. Asymptotically, some terms in a function might become negligible when others become infinitely large or small, which allows for simplification.
- It provides insight into the 'end-behavior' of functions as variables grow.
- It allows comparison of the relative growth rates of different functions or terms.
- Asymptotic analysis leads to identify dominant terms which guide us to make precise predictions about limits.
Infinite Limits
Infinite limits describe the behavior of a function as its input approaches infinity (or negative infinity). When evaluating these limits, we aim to understand how the function behaves, such as whether it approaches a specific value or grows unbounded.
In calculus, finding the infinite limit often involves simplifying the expression using dominant terms. With dominant term simplification, we consider only the terms that most profoundly impact the function’s value at large scales. For instance, in the expression \( \lim_{x \to \infty} \frac{\sqrt{9x^6 - x}}{x^3 + 1} \), we simplify it using the dominant terms for both the numerator and the denominator.
To investigate infinite limits, follow these steps:
In calculus, finding the infinite limit often involves simplifying the expression using dominant terms. With dominant term simplification, we consider only the terms that most profoundly impact the function’s value at large scales. For instance, in the expression \( \lim_{x \to \infty} \frac{\sqrt{9x^6 - x}}{x^3 + 1} \), we simplify it using the dominant terms for both the numerator and the denominator.
To investigate infinite limits, follow these steps:
- Identify and extract dominant terms.
- Simplify the expression using these terms.
- Evaluate the resulting limit expression.
Other exercises in this chapter
Problem 24
(a) If $$G(x)=4 x^{2}-x^{3},\( find$$G^{\prime}(a)\) and use it to find equations of the tangent lines to the curve $$y=4 x^{2}-x^{3}$$ at the points \((2,8)\)
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Evaluate the limit, if it exists. $$\lim _{x \rightarrow-1} \frac{x^{2}+2 x+1}{x^{4}-1}$$
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Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. \(g(\mathrm{x})=\sqrt
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Determine the infinite limit. $$\lim _{x \rightarrow-3^{+}} \frac{x+2}{x+3}$$
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