Problem 12
Question
Find the limits. $$ \lim _{x \rightarrow \infty} \sqrt[3]{\frac{\pi x^{3}+3 x}{\sqrt{2} x^{3}+7 x}} $$
Step-by-Step Solution
Verified Answer
\( \sqrt[3]{\frac{\pi}{\sqrt{2}}} \)
1Step 1: Simplify Inside the Cube Root
We need to find the limit of the expression as \( x \) approaches infinity. So, we start by simplifying the expression inside the cube root:\[\frac{\pi x^3 + 3x}{\sqrt{2} x^3 + 7x}\]. Factor out \( x^3 \) from both the numerator and the denominator:\[\frac{x^3(\pi + \frac{3}{x^2})}{x^3(\sqrt{2} + \frac{7}{x^2})}\].Cancel \( x^3 \) from the numerator and denominator to get:\[\frac{\pi + \frac{3}{x^2}}{\sqrt{2} + \frac{7}{x^2}}\].
2Step 2: Evaluate Limits Inside the Cube Root
Now, evaluate the limit of the simplified expression as \( x \rightarrow \infty \):\[\lim_{x \to \infty} \frac{\pi + \frac{3}{x^2}}{\sqrt{2} + \frac{7}{x^2}}.\]As \( x \rightarrow \infty \), both \( \frac{3}{x^2} \) and \( \frac{7}{x^2} \) approach 0. Thus, \[\lim_{x \to \infty} \frac{\pi + \frac{3}{x^2}}{\sqrt{2} + \frac{7}{x^2}} = \frac{\pi}{\sqrt{2}}.\]
3Step 3: Consider the Cube Root of the Limit
The original expression is the cube root of the fraction we have been evaluating. Therefore, the limit of the original expression is:\[\lim_{x \rightarrow \infty} \sqrt[3]{\frac{\pi x^{3}+3 x}{\sqrt{2} x^{3}+7x}} = \sqrt[3]{\frac{\pi}{\sqrt{2}}}.\]
Key Concepts
Infinite LimitsFactorizationCube RootRational Expressions
Infinite Limits
When dealing with limits as a variable approaches infinity, we often talk about infinite limits. This scenario occurs when we analyze how functions behave as they approach extremely large values. The aim is to determine what value a function converges to, if at all, as a variable grows indefinitely.
In the context of the exercise, as \(x\) tends toward infinity, the terms \(\frac{3}{x^2}\) and \(\frac{7}{x^2}\) shrink towards zero. These terms become negligible, leaving only the dominant terms, \(\pi x^3\) and \(\sqrt{2} x^3\), to guide the overall behavior. Evaluating these dominant terms gives us a clear view of the function's behavior at infinity which is key in finding the limit.
In the context of the exercise, as \(x\) tends toward infinity, the terms \(\frac{3}{x^2}\) and \(\frac{7}{x^2}\) shrink towards zero. These terms become negligible, leaving only the dominant terms, \(\pi x^3\) and \(\sqrt{2} x^3\), to guide the overall behavior. Evaluating these dominant terms gives us a clear view of the function's behavior at infinity which is key in finding the limit.
Factorization
Factorization is a key technique in simplifying expressions, especially when working with rational expressions and approaching limits. It involves breaking down complex expressions into simpler components.
In this particular exercise, by factoring out \(x^3\) from both the numerator and the denominator, we simplify the original rational expression. This technique not only helps in reducing the complexity of the expression but also provides a clearer path to cancel out common terms. Once \(x^3\) is factored out, it becomes easy to see that the algebraically intense terms shrink to zero as \(x\) approaches infinity, revealing the simpler dominant structure \(\frac{\pi}{\sqrt{2}}\).
Factorization is crucial in making limits more approachable and comprehensible.
In this particular exercise, by factoring out \(x^3\) from both the numerator and the denominator, we simplify the original rational expression. This technique not only helps in reducing the complexity of the expression but also provides a clearer path to cancel out common terms. Once \(x^3\) is factored out, it becomes easy to see that the algebraically intense terms shrink to zero as \(x\) approaches infinity, revealing the simpler dominant structure \(\frac{\pi}{\sqrt{2}}\).
Factorization is crucial in making limits more approachable and comprehensible.
Cube Root
The cube root is essential when dealing with expressions that are raised to the power of one-third \((1/3)\). In mathematics, taking a cube root effectively 'undoes' cubing a value, and is particularly useful in equations involving terms raised to the third power.
With the given exercise, once the simplified expression inside the cube root is resolved, finding the cube root of the outcome finalizes the limit computation. In our case, once \(\frac{\pi}{\sqrt{2}}\) is obtained, we further take the cube root: \(\sqrt[3]{\frac{\pi}{\sqrt{2}}}\). This step is central as it directly impacts our answer, illustrating the importance of correctly applying cube roots in limits where they appear.
Understanding cube roots helps us fully process and extract results from complex functions.
With the given exercise, once the simplified expression inside the cube root is resolved, finding the cube root of the outcome finalizes the limit computation. In our case, once \(\frac{\pi}{\sqrt{2}}\) is obtained, we further take the cube root: \(\sqrt[3]{\frac{\pi}{\sqrt{2}}}\). This step is central as it directly impacts our answer, illustrating the importance of correctly applying cube roots in limits where they appear.
Understanding cube roots helps us fully process and extract results from complex functions.
Rational Expressions
Rational expressions are fractions that have polynomials in both the numerator and the denominator. These expressions require careful manipulation to evaluate limits, especially as variables approach infinite values.
In this example, the rational expression \(\frac{\pi x^3 + 3x}{\sqrt{2} x^3 + 7x}\) demonstrates how dominant terms dictate the behavior of the expression as \(x\) becomes very large. Simplification involves canceling out common factors and focusing on higher-degree terms that largely influence the outcome of the limit.
Understanding rational expressions is imperative for comprehending how fractions evolve as variables grow significantly, key to mastering limits at infinity. The exercise highlights the importance of rational expression manipulation by leading to a valid conclusion through systematic simplification.
In this example, the rational expression \(\frac{\pi x^3 + 3x}{\sqrt{2} x^3 + 7x}\) demonstrates how dominant terms dictate the behavior of the expression as \(x\) becomes very large. Simplification involves canceling out common factors and focusing on higher-degree terms that largely influence the outcome of the limit.
Understanding rational expressions is imperative for comprehending how fractions evolve as variables grow significantly, key to mastering limits at infinity. The exercise highlights the importance of rational expression manipulation by leading to a valid conclusion through systematic simplification.
Other exercises in this chapter
Problem 12
, find the indicated limit. In most cases, it will be wise to do some algebra first. $$ \lim _{x \rightarrow 3} \frac{x^{2}-9}{x-3} $$
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Evaluate each limit. $$ \lim _{t \rightarrow 0} \frac{\tan 2 t}{\sin 2 t-1} $$
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State whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ f(t)=\left\\{\begin{array}{ll} t-3 & \text { if } t \leq 3 \
View solution Problem 13
In Problems 13-24, find the indicated limit or state that it does not exist. In many cases, you will want to do some algebra before trying to evaluate the limit
View solution