Problem 270 - Calculus Volume 1

Question

Evaluate the limit. \(\lim _{x \rightarrow \infty} \frac{2 \sqrt{x}}{x-\sqrt{x}+1}\)

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Express with Dominant Terms

When evaluating limits to infinity, identify dominant terms in both the numerator and the denominator. Here, in the numerator, the dominant term is \(2\sqrt{x}\), and in the denominator, it is \(x\). All other terms become negligible as \(x\) approaches infinity.

Step 2: Simplify the Expression

Divide both the numerator and the denominator by \(\sqrt{x}\), the highest power of \(x\) in the numerator. This gives us:\[\frac{2\sqrt{x}}{x - \sqrt{x} + 1} = \frac{2}{\sqrt{x} - 1 + \frac{1}{\sqrt{x}}}\]

Step 3: Evaluate the Simplified Limit

As \(x\) approaches infinity, \(\sqrt{x}\) approaches infinity as well, making \(\frac{1}{\sqrt{x}}\) approach 0. Thus, simplify the expression further:\[\lim_{x \to \infty} \frac{2}{\sqrt{x} - 1 + \frac{1}{\sqrt{x}}} \approx \lim_{x \to \infty} \frac{2}{\sqrt{x}}\]This results in:\[\lim_{x \to \infty} \frac{2}{\sqrt{x}} = 0\]

Step 4: Conclusion: Determine the Final Limit Value

The limit evaluated for \(\frac{2}{\sqrt{x}}\) as \(x\) approaches infinity leads to the final answer which is zero since \(\sqrt{x}\) grows unbounded.

Key Concepts

Dominant Term Analysis

When evaluating limits as a variable approaches infinity, it is important to identify the most influential terms in the expression. These are known as the 'dominant terms'. The dominant terms set the pace of growth or decline in the function's behavior as it approaches infinity. In our exercise, we have an expression in the form of a fraction: \(\frac{2 \sqrt{x}}{x - \sqrt{x} + 1}\). As \(x\) becomes very large, or approaches infinity, we need to find which terms hold the greatest significance.

• Numerator Dominance: The numerator is \(2\sqrt{x}\). It dominates over any constant because as \(x\) becomes very large, its value significantly eclipses other terms.
• Denominator Dominance: In the denominator, the term \(x\) is clearly dominant. It grows faster than \(-\sqrt{x}\) or the constant \(1\).

By understanding these dominant terms, we simplify the original expression, knowing precisely which terms will guide the function's limit behavior.

Simplifying Expressions

Simplifying expressions is a crucial step in evaluating limits. It involves reducing a complex expression to its simplest form, focusing on the dominant terms identified in the previous step. In this exercise, simplifying involves dividing both the numerator and the denominator by \(\sqrt{x}\), the term that balances both the components involved:\[\frac{2\sqrt{x}}{x - \sqrt{x} + 1} = \frac{2}{\sqrt{x} - 1 + \frac{1}{\sqrt{x}}}\]Once this is accomplished, the expression is much easier to handle for the final infinite limit calculation.

• The new expression makes clear the diminishing impact of terms like \(-1\) and \(\frac{1}{\sqrt{x}}\) as \(x\) increases.
• This simplification aligns all terms for streamlined limit evaluation.

Thus, simplifying expressions allows us to directly focus on the most significant behavior of the function as \(x\) approaches infinity.

Infinity Limits

Understanding infinity limits is key to mastering limits involving \( x \) tending towards infinity. It often requires assessing how expressions behave when their variable becomes unbounded. In our exercise, we simplify the expression \[\lim_{x \to \infty} \frac{2}{\sqrt{x} - 1 + \frac{1}{\sqrt{x}}}\]As \(x\) increases indefinitely, we note important trends:

• The denominator's dominant term, \(\sqrt{x}\), grows larger, meaning that any impact from \(-1\) and \(\frac{1}{\sqrt{x}}\) diminishes progressively.
• As a result, the expression approaching \(\infty\) is further simplified to \(\frac{2}{\sqrt{x}}\).

This form confirms that \[\lim_{x \to \infty} \frac{2}{\sqrt{x}} = 0\]The closer \(\sqrt{x}\) gets to infinity, the closer the expression \(\frac{2}{\sqrt{x}}\) approaches zero.Understanding how to handle limits this way simplifies calculations and ensures an accurate assessment of the behavior of functions at the extremes of their domain.