Problem 47 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(45-48\) , 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. $$ \lim _{x \rightarrow-\infty}\left(3 x+\sqrt{9 x^{2}-x}\right) $$

Step-by-Step Solution

Answer

The limit as x approaches -∞ of the expression \(3x + \sqrt{9x^{2} - x}\) is 9.

Step 1: Rationalize the Numerator

To rationalize the numerator in this case means to eliminate the square root from the numerator of the fraction. The hint suggests treating the whole expression as a fraction with denominator 1. This can be accomplished by multiplying the expression by the conjugate of the expression under the square root. The conjugate of \(9x^{2} - x\) is \(9x^{2} + x\). Thus, the expression becomes \[ \frac{(3x + \sqrt{9x^{2} - x})(3x - \sqrt{9x^{2} + x})}{1 * (3x - \sqrt{9x^{2} + x})}\].

Step 2: Simplify the Expression

Expanding and simplifying the numerator yields \(9x^{2} - x\), and this can be further simplified by dividing each term in the expression by x. Since the limit is as x approaches -∞, the lower degree term -x/x = -1, can be neglected to determine the limit. Thus, the simplified expression becomes \(9-\frac{1}{x}\). The new limit is now \[ \lim_{x \rightarrow -\infty} (9-\frac{1}{x})\].

Step 3: Find the Limit

Taking the limit as x goes to -∞ of the simplified expression gives \(9 - 0 = 9\), since any expression in the form of \(\frac{a}{x}\) where a is a finite number and x approaches +/-∞ always equals 0.

Step 4: Verify the Result Using a Graphing Utility

Use a graphing utility, such as a graphing calculator or online graphing tool, to graph the original function. As x approaches -∞, check to see if the y-values approach 9 to confirm that the limit appears to be correct.

Key Concepts

Rationalizing Expressions

Rationalizing expressions is an important technique in calculus. It helps simplify complex expressions that involve square roots or other irrational components. In this exercise, we needed to rationalize the numerator of the expression \(3x + \sqrt{9x^{2} - x}\). To do this effectively:

• Identify the component involving the square root. Here, it's \(\sqrt{9x^{2} - x}\).
• Multiply both the numerator and denominator by the conjugate of this component. For the numerator \(\sqrt{9x^{2} - x}\), the conjugate is \(\sqrt{9x^{2} + x}\).

The purpose of this multiplication is to eliminate the square root, leaving a simpler polynomial expression. After multiplying and simplifying, the original complex expression becomes easier to handle.

Graphical Verification of Limits

Visualizing limits helps us understand their behavior beyond algebraic calculations. In this case, once we simplified the expression and found the limit, we used a graphing utility to verify our solution. Graphing the expression \(3x + \sqrt{9x^{2} - x}\) as \(x\) approaches \(-\infty\) provides a visual confirmation:

• Plot the function using a graphing calculator or online tool.
• Observe the values of \(y\) as \(x\) moves towards \(-\infty\).
• Check if the graph approaches the limit found algebraically, here 9.

By confirming the work graphically, students gain confidence in the validity of their solved limits, bridging the gap between abstract calculations and concrete visual results.

Infinity in Calculus

Infinity plays a unique role in calculus, allowing us to explore behavior of functions as variables grow extremely large or small. In the provided exercise, we approached the limit as \(x\) tends to \(-\infty\). Here are key steps involved:

• Understand that \(-\infty\) represents a direction rather than a number. It indicates moving leftward indefinitely on the number line.
• Utilize algebraic simplifications like removing fractions where the denominator grows indefinitely, (e.g., \(\frac{1}{x} \rightarrow 0\) as \(x \rightarrow -\infty\)).
• Conclude the limit of the expression as it moves towards this extreme value, knowing that small changes become negligible compared to such large scales.

The exploration of infinity provides deep insights into how functions behave at extreme ranges, crucial in understanding limits and continuity in calculus.