Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{x \rightarrow \pi / 4} \frac{(x-\pi / 4)^{2}}{(\tan x-1)^{2}} $$

Step-by-Step Solution

Answer

The limit is 1. Use repeated L'Hôpital's Rule and graphical observation.

Step 1: Understand the Limit

We need to calculate the limit, \( \lim_{x \rightarrow \pi / 4} \frac{(x-\pi / 4)^{2}}{(\tan x-1)^{2}} \). This expression is in the form \( \frac{0}{0} \) when \( x = \pi / 4 \), which suggests we need to use L'Hôpital's Rule.

Step 2: Apply L'Hôpital's Rule

L'Hôpital's Rule states that if \( \lim_{x \rightarrow a} \frac{f(x)}{g(x)} = \frac{0}{0} \), then \( \lim_{x \rightarrow a} \frac{f(x)}{g(x)} = \lim_{x \rightarrow a} \frac{f'(x)}{g'(x)} \). In our case, differentiate the numerator and the denominator with respect to \( x \):\[ f'(x) = 2(x - \frac{\pi}{4}) \text{ and } g'(x) = 2(\tan x - 1)\sec^2 x \].

Step 3: Compute the Derivatives

Using the derivatives from step 2, substitute them back into the limit:\[ \lim_{x \rightarrow \pi / 4} \frac{2(x - \frac{\pi}{4})}{2(\tan x - 1) \sec^2 x} \].This can simplify to:\[ \lim_{x \rightarrow \pi / 4} \frac{x - \frac{\pi}{4}}{(\tan x - 1) \sec^2 x} \].

Step 4: Evaluate the Limit Again

We again have the \( \frac{0}{0} \) form. Re-apply L'Hôpital's Rule:Derivatives are \( 2 \) for the numerator and after further simplification, \[ 2\sec^2 x \cdot (\sec^2 x + 2 \tan x) \] for the denominator.Re-evaluate now for \( x = \pi/4 \).

Step 5: Numerical Computation Using a Calculator

After simplifying and using L'Hôpital's Rule repeatedly, calculate this limit numerically with a graphing calculator and observe the behavior near \( x = \pi/4 \). This evaluation will give you the value of the limit.

Step 6: Plot the Function Graphically

Use a graphing calculator to plot \( y = \frac{(x-\pi / 4)^{2}}{(\tan x-1)^{2}} \) around \( x = \pi/4 \). Observe the behavior of the function as \( x \) approaches \( \pi/4 \) from both left and right sides.

Key Concepts

L'Hôpital's Rule

When faced with a limit problem that results in an indeterminate form like \( \frac{0}{0} \), L'Hôpital's Rule is a valuable tool. This rule allows us to differentiate the numerator and denominator separately and then take the limit again. It simplifies the process of finding limits of fractions when direct substitution gives indeterminate forms.
Think of it as a shortcut through differentiation, pathing a clear route out of the confusion.
To apply L'Hôpital's Rule, make sure:

• Your limit initially gives \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) when directly substituted.
• Differentiate the numerator and the denominator separately.
• Re-evaluate the limit with the new expressions.
• If you end up with another indeterminate form, apply the rule again.

This technique redeems a seemingly troublesome limit into a more manageable calculation, saving time and effort.

Graphing Calculator

Graphing calculators are powerful tools that visualize functions and aid in understanding mathematical concepts like limits. By graphing the function \( y = \frac{(x-\pi / 4)^{2}}{(\tan x-1)^{2}} \), you can observe the behavior of the function near the point \( x = \pi / 4 \).Here's how you can leverage a graphing calculator:

• Enter the function into the graphing interface.
• Set a suitable window that focuses on \( x = \pi / 4 \).
• Zoom in closely to see how the function behaves as \( x \) approaches the point from the left and right.
• This visual approach complements numerical methods, providing a holistic view.

The graphical representation will not only show where the function may potentially have a limit but also help you identify any discontinuities or peculiar behaviors.

Numerical Computation

In addition to analytical solutions, numerical computation provides a practical way to estimate limits using a calculator. It involves calculating the function values closer and closer to the limit point to identify trends or approaches.Here's a simple way to carry out numerical computation:

• Choose values of \( x \) that are incrementally closer to \( \pi/4 \).
• Calculate the corresponding function values \( y \).
• Analyze how \( y \) changes as \( x \) hones in on \( \pi/4 \).

Through this method, you can numerically verify the analytical limit or investigate the function's behavior when other methods are cumbersome. This hands-on approach also paves the way for stronger intuition about limits.

Continuity and Differentiability

Continuity and differentiability are foundational concepts in calculus, closely tied to the behavior of limits. A function is continuous at a point if it is defined, its limit exists as the point is approached, and the function's value equals the limit. Differentiability steps it up a notch, requiring the existence of a derivative at that point.For example, for the given function, you explore its continuity at \( x = \pi/4 \) by:

• Ensuring it is defined around this point.
• Checking if the limit as \( x \to \pi/4 \) equals the function's value at \( \pi/4 \).

Differentiability implies continuity, but the reverse is not guaranteed; some continuous functions may not be differentiable.
Understanding these qualities helps you predict and explain how functions will behave and change, which is crucial for solving limits.