Question

\(\lim _{x \rightarrow \infty} \frac{2 \cdot x^{1 / 2}+3 \cdot x^{1 / 3}+4 \cdot x^{1 / 4}+\ldots . .+n \cdot x^{1 / n}}{(3 x-4)^{1 / 2}+(3 x-4)^{1 / 3}+\ldots+(3 x-4)^{1 / n}}(\) here \(n \in N, n \geq 2)\) is equal to (1) \(\frac{2}{\sqrt{3}}\) (2) \(\frac{\sqrt{3}}{2}\) (3) \(\frac{1}{2}\) (4) \(\frac{1}{\sqrt{3}}\)

Short answer

The limit is \( \frac{2}{\sqrt{3}} \), which corresponds to option (1).

Step-by-step solution

Analyze the problem

Given the limit \(\lim _{x \rightarrow fty} \frac{2 \cdot x^{1 / 2}+ 3 \cdot x^{1 / 3}+ 4\cdot x^{1 / 4}+\ldots\cdot n \cdot x^{1 / n}}{(3 x-4)^{1 / 2}+(3 x-4)^{1 / 3}+\ldots+(3 x-4)^{1 / n}}\) where \( n \in \mathbb{N}, \ n \geq 2 \). The goal is to find the limit as \( x \) approaches infinity.

Simplify the numerator and denominator for large values of x

For large values of \( x \, x^{1 / 2} \) will dominate over terms like \( x^{1 / 3}, x^{1 / 4}, ... \), so we focus on the highest exponent term. Hence, in the numerator, \( 2 \cdot x^{1/2} \) will be the leading term. Similarly, in the denominator, \( (3x-4)^{1/2} \) will dominate.

Approximate the dominant terms

For simplification, approximate \( (3x - 4)^{1/2} \approx 3^{1/2} \cdot x^{1/2} \) when \( x \) is very large. Thus, the problem reduces to: \lim _{x \rightarrow \infty} \frac{2 \cdot x^{1 / 2}}{3^{1/2} \cdot x^{1 / 2}}\.

Cancel common terms and find the limit

Now, cancel the common term \( x^{1 / 2} \) in the numerator and denominator: \( \lim _{x \rightarrow \infty} \frac{2 \cdot x^{1 / 2}}{3^{1 / 2} \cdot x^{1 / 2}} = \frac{2}{3^{1/2}} = \frac{2}{\sqrt{3}}.\)

Key concepts

Limits and Continuity

Limits are vital in understanding the behavior of functions as they approach a certain point, either a finite value or infinity. In this exercise, we dealt with a limit as the variable x approaches infinity. This helps us see how the function behaves at very large values.
Continuity is closely related to limits. A function is continuous at a point if the limit exists there and matches the function’s actual value. For our problem, continuity ensures the function doesn’t have any jumps or breaks.
The idea of limits provides a foundation for calculus, helping us understand derivatives and integrals. Practicing these problems enhances our ability to think about how functions change and behave, not just at specific points but overall.

Asymptotic Behavior

Asymptotic behavior looks at how functions behave as the input values become very large. In our problem, we investigated the function as x approached infinity.
Here, the numerator and the denominator had terms like \(x^{1/2}\), \(x^{1/3}\), etc. As x becomes very large, the dominant term among these has the highest power, which in our case was \(x^{1/2}\). Understanding which terms dominate helps simplify complex expressions.
The term 'asymptote' refers to a line that a graph approaches but never actually reaches. While this term wasn’t directly used in the problem, recognizing dominant terms is key to understanding asymptotic behavior.

Dominant Terms in Limits

Dominant terms are those that have the most influence on the value of an expression as variables approach a limit, either because they grow fastest or shrink slowest.
In our limit problem, as x approaches infinity, \(x^{1/2}\) dominates over lesser powers like \(x^{1/3}\) and \(x^{1/4}\). This understanding let us focus only on these dominant terms to simplify our problem.
In the solution, we looked at the numerator and denominator separately. We identified \(2 \cdot \ x^{1/2}\) in the numerator and \( (3x - 4)^{1/2} \) in the denominator as dominant terms.
By approximating the effect of these terms, we made the problem manageable. This technique is particularly useful in calculus for solving limits and can be widely applied to different problems.