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Magazine Chapter 8: Problem 38
Compute the limits. $$ \lim _{x \rightarrow \infty} \frac{4 x}{\sqrt{2 x^{2}+1}} $$
Short Answer
Expert verified
The answer to the limit is \( 2 + \sqrt{2} \).
Step by step solution
01
Rationalize the denominator
Firstly, the function needs to be simplified. This can be done by multiplying the numerator and the denominator by the conjugate of the denominator. So, \( \frac{4x}{\sqrt{2x^{2}+1}} \) becomes \( \frac{4x(\sqrt{2x^{2}+1})}{\sqrt{2x^{2}+1}(\sqrt{2x^{2}+1})} \) which further simplifies into \( \frac{4x\sqrt{2x^{2}+1}}{2x^{2}+1} \).
02
Separate the functions in the numerator
The next step is to separate the function in the numerator to make simplification easier. Therefore, \( \frac{4x\sqrt{2x^{2}+1}}{2x^{2}+1} \) becomes \( \frac{4x}{2x^{2}+1} + \frac{\sqrt{2x^{2}+1}}{2x^{2}+1} \).
03
Apply limit rules
Now we can apply the limit rules. Divide every term in the numerator and the denominator by \(x^{2}\) to simplify the expression and then apply the limit. The first term transforms to \( \frac{4}{2+\frac{1}{x^{2}}} \) and the second term becomes \( \frac{\sqrt{2+\frac{1}{x^{2}}}}{2+\frac{1}{x^{2}}} \). As \(x\) approaches infinity, \( \frac{1}{x^{2}} \) approaches 0. Therefore, the limits of the two terms are 2 and \( \sqrt{2} \) respectively.
04
Adding the limits
The limit of the sum is the sum of the limits, therefore add the two limits together. Here, it will be 2 + \( \sqrt{2} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Infinite Limits
When dealing with infinite limits, we explore the behavior of a function as the input, typically denoted as \(x\), grows without bound. For instance, in the exercise, we computed the limit of a rational expression as \(x\) approached infinity.
To understand this, consider that as \(x\) increases, certain terms become negligible in comparison to others. Focus on the terms that grow fastest in both the numerator and denominator.
To understand this, consider that as \(x\) increases, certain terms become negligible in comparison to others. Focus on the terms that grow fastest in both the numerator and denominator.
- If a function grows faster in the denominator, the limit generally tends toward zero.
- Conversely, if the numerator's growth is dominant, the limit tends toward infinity.
- When both grow at similar rates, division of their leading coefficients typically dictates the limit.
Rationalization
Rationalization is a technique often used to simplify complex expressions by eliminating square roots or radicals from the denominator. This is particularly useful when dealing with limits, as it makes the expression more manageable.
In the exercise, the denominator \(\sqrt{2x^{2}+1}\) was rationalized by multiplying both the numerator and the denominator by itself. This process transforms the expression into a simpler rational function.
By rationalizing:
In the exercise, the denominator \(\sqrt{2x^{2}+1}\) was rationalized by multiplying both the numerator and the denominator by itself. This process transforms the expression into a simpler rational function.
By rationalizing:
- We eliminate the square root, leading to an expression that is easier to evaluate, especially when applying limits.
- Simplifies future calculations, since handling radicals can complicate arithmetic operations.
Limit Rules
Limit rules are foundational tools in calculus that allow us to find limits of functions efficiently. Some of the most applicable rules in this context include:
In the solution, these rules allowed us to simplify the expression by dividing each term in the numerator and the denominator by the highest power of \(x\). Consequently, it becomes increasingly easier to determine the behavior of each separate part as \(x\) tends to infinity.
Understanding and applying these rules simplify complex problems into manageable parts, making them essential for limit calculus.
- Quotient Rule: The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero.
- Constant Multiple Rule: A constant factor can be pulled out in front of the limit sign.
- Sum or Difference Rule: The limit of a sum or difference is the corresponding sum or difference of their limits.
In the solution, these rules allowed us to simplify the expression by dividing each term in the numerator and the denominator by the highest power of \(x\). Consequently, it becomes increasingly easier to determine the behavior of each separate part as \(x\) tends to infinity.
Understanding and applying these rules simplify complex problems into manageable parts, making them essential for limit calculus.
Simplifying Expressions
Simplifying expressions is a crucial skill in mathematics, helping to reduce complexity and make evaluating limits more straightforward. The goal is to transform the expression into a simpler form, often by:
In the context of infinity, we focus on isolating the terms that dominate growth. For instance, terms like \(\frac{1}{x^2}\) fall to zero, streamlining the expression and illuminating the path to solving the limit.
This reduction transforms an initially daunting equation into a much more approachable problem, highlighting the beauty of calculus and its ability to handle infinite behavior seamlessly.
- Combining like terms and reducing fractions.
- Rationalizing denominators, as seen in our exercise.
- Dividing terms to eliminate smaller ones that become insignificant as \(x\) approaches large values.
In the context of infinity, we focus on isolating the terms that dominate growth. For instance, terms like \(\frac{1}{x^2}\) fall to zero, streamlining the expression and illuminating the path to solving the limit.
This reduction transforms an initially daunting equation into a much more approachable problem, highlighting the beauty of calculus and its ability to handle infinite behavior seamlessly.
