Problem 48 - 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. $$ \lim _{x \rightarrow-\infty}\left(\frac{x}{2}+\sqrt{\frac{1}{4} x^{2}+x}\right) $$

Step-by-Step Solution

Answer

The limit is \(-\infty\).

Step 1: Rationalize the numerator

Multiply the given expression by its conjugate separated by 1. The conjugate of the given expression \(\frac{x}{2}+\sqrt{\frac{1}{4}x^{2}+x}\) is \(\frac{x}{2}-\sqrt{\frac{1}{4}x^{2}+x}\). After this, simplify the expression.

Step 2: Simplify the obtained expression

After simplifying, the expression becomes \(-\sqrt{\frac{1}{4}x^{2}+x}\).

Step 3: Evaluate the limit

After having simplified the expression, the limit can be calculated directly: \(\lim _{x \rightarrow-\infty}[-\sqrt{\frac{1}{4}x^{2}+x}] = \lim _{x \rightarrow-\infty}-\sqrt{\frac{1}{4}x^{2}+x}\). Since x tends to \(-\infty\), we get \(-\infty\) as the output.

Key Concepts

Rationalization Technique

The rationalization technique is an important tool in calculus, particularly when dealing with expressions that include square roots or other irrational components. It involves multiplying the expression by a carefully chosen form of 1. This form is usually the conjugate of the irrational part of the expression. The goal is to eliminate the radical from the expression. In this exercise, we start with the function \(\frac{x}{2} + \sqrt{\frac{1}{4}x^2 + x}\). Here, the irrational part is the square root. Hence, we multiply it by its conjugate, \(\frac{x}{2} - \sqrt{\frac{1}{4}x^2 + x}\), over itself to maintain the equality.

• *Step 1:* Write the expression with its conjugate as a fraction and multiply them.
• *Step 2:* Simplify the numerator by applying the difference of squares formula \((a - b)(a + b) = a^2 - b^2)\).

This results in a form where the numerator no longer has a square root, making further simplification possible. This technique is crucial for handling expressions as we approach limits.

Infinite Limits

Infinite limits provide insight into the behavior of functions as they approach a very large (infinite) positive or negative value. In calculus, one examines how the function behaves as \(x\) approaches \(\infty\) or \(-\infty\). For the given function \(\lim_{x \rightarrow -\infty}\left(\frac{x}{2} + \sqrt{\frac{1}{4}x^{2} + x}\right)\), understanding infinite limits helps predict how the function behaves as \(x\) becomes increasingly negative. In this solution, once we rationalize the expression and simplify, the limit becomes easier to evaluate:

• First, observe that the simplified expression is \(-\sqrt{\frac{1}{4}x^2 + x}\).
• Since the dominant term \(\frac{1}{4}x^{2}\) is positive and \(x\) becomes very large and negative, the overall expression trends toward \(-\infty\).

Infinite limits are a powerful concept for understanding asymptotic behavior and trends of functions at extreme ends.

Graphical Verification

Graphical verification is a practical step that complements algebraic solutions. By graphing a function, students can visually confirm the behavior of the function, such as its trend as \(x\) approaches infinity. In this exercise, using a graphing utility to plot \(\frac{x}{2} + \sqrt{\frac{1}{4}x^2 + x}\) reveals how the function behaves as \(x\) approaches \(-\infty\). The graph will show:

• A steep decline, illustrating the limit that approaches \(-\infty\).
• The continuous downward motion, confirming consistency with algebraic findings.

Graphical verification serves as an invaluable tool for reinforcing one’s understanding and verifying algebraic calculations. Visual aids can clarify complex concepts and enhance learning by providing an intuitive grasp of the behavior of functions.