Problem 72 - Thomas Calculus

Question

The asymptotes of \(\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1\) Show that the vertical distance between the line \(y=(b / a) x\) and the upper half of the right-hand branch \(y=(b / a) \sqrt{x^{2}-a^{2}}\) of the hyperbola \(\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1\) approaches 0 by showing that $$ \lim _{x \rightarrow \infty}\left(\frac{b}{a} x-\frac{b}{a} \sqrt{x^{2}-a^{2}}\right)=\frac{b}{a} \lim _{x \rightarrow \infty}\left(x-\sqrt{x^{2}-a^{2}}\right)=0 $$ Similar results hold for the remaining portions of the hyperbola and the lines \(y=\pm(b / a) x .\)

Step-by-Step Solution

Answer

The vertical distance approaches 0 as \( x \rightarrow \infty \).

Step 1: Set up the problem

We need to examine the vertical distance between the line \( y = \frac{b}{a} x \) and the branch of the hyperbola given by \( y = \frac{b}{a} \sqrt{x^2 - a^2} \). This distance is represented by \( \frac{b}{a} x - \frac{b}{a} \sqrt{x^2 - a^2} \). We will calculate the limit of this expression as \( x \rightarrow \infty \).

Step 2: Simplify the expression

Factor out \( \frac{b}{a} \) from the expression:\[\frac{b}{a} \left( x - \sqrt{x^2 - a^2} \right)\] We will focus on evaluating the limit of \( x - \sqrt{x^2 - a^2} \) as \( x \) approaches infinity.

Step 3: Rewrite the expression inside the limit

To simplify \( x - \sqrt{x^2 - a^2} \), multiply and divide by the conjugate:\[\frac{(x - \sqrt{x^2 - a^2})(x + \sqrt{x^2 - a^2})}{x + \sqrt{x^2 - a^2}}\] This simplifies the numerator to:\[ x^2 - (x^2 - a^2) = a^2 \] Thus, the expression becomes:\[ \frac{a^2}{x + \sqrt{x^2 - a^2}} \]

Step 4: Evaluate the limit

Now, calculate the limit:\[ \lim_{x \rightarrow \infty} \frac{a^2}{x + \sqrt{x^2 - a^2}} \]As \( x \rightarrow \infty \), both \( x \) and \( \sqrt{x^2 - a^2} \) approach \( x \) in growth rate, making the denominator approximately \( 2x \). Thus:\[\lim_{x \rightarrow \infty} \frac{a^2}{x + \sqrt{x^2 - a^2}} = 0\]

Step 5: Apply the result to the original problem

Substitute the evaluated limit back:\[\frac{b}{a} \lim_{x \rightarrow \infty} \left( x - \sqrt{x^2 - a^2} \right) = \frac{b}{a} \cdot 0 = 0\]This confirms the original query that the vertical distance approaches 0.

Key Concepts

Understanding Asymptotes

The concept of asymptotes is fundamental when discussing hyperbolas. An asymptote is a line that a curve approaches as it heads towards infinity. For the hyperbola given by the equation \[\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1\]the asymptotes are lines that the hyperbola approaches but never actually reaches. In our specific context, the asymptotes are represented by the lines \( y = \pm \frac{b}{a}x \).

Here, these asymptotes serve as boundary lines that define how the branches of the hyperbola behave as they extend further into the plane. They are vital for understanding the long-term behavior of the curves, especially when calculating limits.

In analyzing the vertical distance between the hyperbola and its asymptote, we emphasize the role of the asymptotes as reference guides, ensuring that the hyperbola remains within these bands as it spreads out.

Exploring Limits

In mathematics, limits allow us to understand how a function behaves as it approaches a certain point or infinity. Here, we use limits to determine the behavior of the vertical distance between the hyperbola's branch and its asymptote as \( x \) becomes very large.

The expression we evaluate is:\[\lim_{x \rightarrow \infty} \left( \frac{b}{a} x - \frac{b}{a} \sqrt{x^2 - a^2} \right)\]By examining this limit, we ascertain that the difference becomes negligible. This essentially means that the hyperbola's branch nears the asymptote progressively as \( x \) increases.

Understanding limits help us grasp the essential "end behavior" of the hyperbola, illustrating how it's bound by its asymptotes, further justifying why the vertical distance approaches zero for large values of \( x \).

Conjugate Expression Usage

To simplify the expression \( x - \sqrt{x^2 - a^2} \), we utilize the conjugate expression. The technique involves multiplying by the conjugate, \( x + \sqrt{x^2 - a^2} \), over itself:\[\frac{(x - \sqrt{x^2 - a^2})(x + \sqrt{x^2 - a^2})}{x + \sqrt{x^2 - a^2}}\]

This step simplifies the expression significantly. By doing so, the numerator simplifies to:\[a^2\]The simplification uses the difference of squares formula, helping reduce complexity and allowing easier computation of limits.

• Multiplies by conjugate to rationalize expressions.
• Facilitates easier limit calculations.

This insightful trick highlights the elegance often found in mathematical manipulation, turning a seemingly complex problem into a manageable context.

Evaluating Vertical Distance

The vertical distance between a curve and its asymptote provides insight into how closely they align. For the given hyperbola, the vertical distance is determined by:\[\frac{b}{a} x - \frac{b}{a} \sqrt{x^2 - a^2}\]The problem asks us to show that this distance approaches zero as \( x \) becomes very large. The solution involves calculating the limit of this difference.

With the simplification using the conjugate, we find:\[\lim_{x \rightarrow \infty} \frac{a^2}{x + \sqrt{x^2 - a^2}} = 0\]Thus, this confirms that as you look further into the horizon of the graph, the vertical distance shrinks closer to zero.

• Ensures the hyperbola aligns closely with its asymptote.
• Shows the significance of limits in verifying asymptotic behavior.

In essence, this process corroborates the understanding that asymptotes act as a guiding light for the graph as it stretches outward.