Problem 44 - Essential Calculus

Question

In Exercises \(43-48,\) find the limit. (Hint: Treat the expression as a fraction whose denominator is 1 , and rationalize the numerator.) Use a graphing utility to verify your result. $$ \lim _{x \rightarrow \infty}\left(2 x-\sqrt{4 x^{2}+1}\right) $$

Step-by-Step Solution

Answer

The limit of the function \(2x - \sqrt{4x^2 + 1}\) as \(x\) approaches infinity is 0.

Step 1: Rewrite the Expression

First, rewrite the expression as a fraction \(\frac{2x - \sqrt{4x^2 + 1}}{1}\) to set up for rationalization.

Step 2: Rationalize the Numerator

Rationalize the numerator by multiplying the whole expression by \(\frac{2x + \sqrt{4x^2 + 1}}{2x + \sqrt{4x^2 + 1}}\). This gives us \(\lim _{x \rightarrow \infty}\frac{4x^2 - \left(4x^2 + 1\right)}{2x + \sqrt{4x^2 + 1}}.\) This simplifies to \(\lim _{x \rightarrow \infty}\frac{-1}{2x + \sqrt{4x^2 + 1}}.\)

Step 3: Substitute \(x\) with Infinity

To find the limit as \(x\) approaches infinity, replace \(x\) with infinity in \(\frac{-1}{2x + \sqrt{4x^2 + 1}}\). Considering that the square root term dominates when \(x\) is very large, the expression simplifies to \(\frac{-1}{2\infty + \infty}\) which is \(\frac{-1}{\infty}\)

Step 4: Evaluate the Limit

The limit of a constant divided by infinity is 0. So, \(\frac{-1}{\infty} = 0\)

Key Concepts

Rationalization

Rationalization is a helpful algebraic technique when you want to simplify expressions, especially those involving radicals. By rationalizing the expression, you can transform a fraction with complicated expressions in the numerator or denominator, which makes finding limits or derivatives easier.
To rationalize, multiply both the numerator and denominator by the conjugate of either. In this particular example, when dealing with \(2x - \sqrt{4x^2 + 1}\), the conjugate \(2x + \sqrt{4x^2 + 1}\) was used. This effectually removes the square root from the numerator as shown in: \((2x - \sqrt{4x^2 + 1})\) becomes \(\frac{4x^2 - (4x^2 + 1)}{2x + \sqrt{4x^2 + 1}}\).
The benefit of rationalization includes:

• Eliminating complex radicals
• Facilitating limit evaluation
• Simplifying algebraic expressions for easier computation

Rationalization is a powerful trick, especially in calculus. It frequently simplifies expressions that might seem complex at first glance.

Infinite Limits

Infinite limits are a fundamental concept in calculus, involving evaluating what happens to a function as the variable approaches infinity or negative infinity. We consider how terms grow or shrink as they approach extremely large values.
In the example, we have \lim _{x \rightarrow \infty} \left(2x - \sqrt{4x^2 + 1}\right).\ As \(x\) approaches infinity, the dominant term is \(\sqrt{4x^2 + 1}\) since it's inside a square root reflecting continuous growth. After rationalization, the simplified form was \frac{-1}{2x + \sqrt{4x^2 + 1}}\, showing that as \(x\) grows, the denominator also grows, leading to a limit value of 0.
This teaches us that when evaluating infinite limits:

• Focus on dominant terms that significantly impact the limit result
• Simplify fractions to reveal how they behave as variables increase
• Understand that dividing by infinitely large quantities results in a value approaching zero

Learning about infinite limits helps in grasping how functions behave at their extremes, an essential skill for predicting behavior and testing hypotheses in advanced math and science.

Graphing Utilities

Graphing utilities are digital tools that allow you to visualize functions and limits. They are invaluable for checking mathematical work, especially complex calculus problems.
By inputting the expression \(2x - \sqrt{4x^2 + 1}\) into a graphing calculator, you can see the behavior of the function as \(x\) approaches infinity. The graph would confirm that the function's output approaches zero, validating the analytical findings from the limits calculation.
Benefits of using graphing utilities:

• Visualize expressions and their limits quickly
• Check the reasoning of manual calculations
• Explore how modifications in expression affect results

While theoretical mathematics provides the why, graphing utilities offer intuitive understanding through real-time visualization, helping strengthen your grasp of complex calculus concepts.