Problem 59 - Precalculus

Question

Evaluate the following limits using a table of values. Given \(f(x)=\left\\{\begin{array}{ll}\sin x & x<\frac{\pi}{4} \\ \tan x & x=\frac{\pi}{4}, \text { find } \\ \cos x & x>\frac{\pi}{4}\end{array}\right.\) a. $$\lim _{x \rightarrow \frac{\pi}{4}} f(x)$$ b. $$\lim _{x \rightarrow \frac{\pi}{4}+} f(x)$$ c. $$\lim _{x \rightarrow \frac{\pi}{4}} f(x)$$

Step-by-Step Solution

Answer

Limits from both directions are \(\frac{\sqrt{2}}{2}\), but the limit doesn't exist at \(x = \frac{\pi}{4}\).

Step 1: Understanding the Function

The piecewise function is defined as follows: - For \(x < \frac{\pi}{4}\), \(f(x) = \sin x\).- At \(x = \frac{\pi}{4}\), \(f(x) = \tan x\).- For \(x > \frac{\pi}{4}\), \(f(x) = \cos x\). Let’s evaluate the limits from different approaches to determine the behavior of \(f(x)\) near \(x = \frac{\pi}{4}\).

Step 2: Evaluate the Left-Hand Limit

To find \(\lim_{x \to \frac{\pi}{4}^{-}} f(x)\), consider values of \(x\) slightly less than \(\frac{\pi}{4}\). Here we use \(f(x) = \sin x\). Calculate \(\sin\left(\frac{\pi}{4} - 0.01\right)\), \(\sin\left(\frac{\pi}{4} - 0.001\right)\), etc. This leads you to approximate the limit as \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\).

Step 3: Evaluate the Right-Hand Limit

To evaluate \(\lim_{x \to \frac{\pi}{4}^{+}} f(x),\) consider values of \(x\) slightly more than \(\frac{\pi}{4}\). Here \(f(x) = \cos x\). Calculate \(\cos\left(\frac{\pi}{4} + 0.01\right)\), \(\cos\left(\frac{\pi}{4} + 0.001\right)\), etc. This also leads you to \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\).

Step 4: Check the Limit at \(x = \frac{\pi}{4}\)

At \(x = \frac{\pi}{4}\), \(f(x) = \tan x\). Calculate \(\tan\left(\frac{\pi}{4}\right)\). We know \(\tan\left(\frac{\pi}{4}\right) = 1\).

Step 5: Conclusion on Limits

For \(\lim_{x \to \frac{\pi}{4}} f(x)\), the left-hand and right-hand limits are both \(\frac{\sqrt{2}}{2}\), but do not match with the function value at \(\frac{\pi}{4}\) which is \(1\).Therefore, the overall limit \(\lim_{x \to \frac{\pi}{4}} f(x)\) does not exist due to the discontinuity at the point.

Key Concepts

Piecewise Functions

A piecewise function is a type of function that has different expressions or rules for various intervals of its domain. This kind of function allows for changing behavior at certain points. In the specified exercise, the function is defined with three distinct parts:

• For values of \(x\) less than \(\frac{\pi}{4}\), the function is given by \(f(x) = \sin x\).
• At exactly \(x = \frac{\pi}{4}\), the function switches to \(f(x) = \tan x\).
• For \(x\) greater than \(\frac{\pi}{4}\), it becomes \(f(x) = \cos x\).

This setup shows how a piecewise function can seamlessly combine different equations within a single mathematical context. Understanding the precise intervals and their corresponding expressions is key to dealing with these functions effectively. Each segment of the interval needs careful consideration when evaluating limits.

Trigonometric Limits

In calculus, trigonometric limits explore the behavior of trigonometric functions as the input variable approaches a particular point. For the piecewise function in the exercise, we must deal with trigonometrics when assessing limits:

• The use of \(\sin x\) when approaching \(\frac{\pi}{4}\) from the left involves examining how the sine function behaves as \(x\) gets closer and closer.
• Dominating as we approach from the right is \(\cos x\), which similarly provides insights when \(x\) nears \(\frac{\pi}{4}\) just from the opposite direction.
• The evaluation at \(x = \frac{\pi}{4}\) itself requires \(\tan x\), whose specific value at this ideal point can give real insight into the function's peculiarity.

Understanding how each function behaves near a critical point allows one to conclude about the nature of limits at that point. These evaluations are fundamental in mastering the composition and limits of trigonometric functions.

Left-Hand Limit

Left-hand limits involve evaluating the behavior of a function as the variable approaches a particular value from the left. Symbolically, it is represented as \(\lim_{x \to c^-} f(x)\), where \(c\) is the point being approached. For our exercise, the left-hand limit of \(f(x)\) as \(x\) approaches \(\frac{\pi}{4}\) is computed with \(f(x) = \sin x\).To evaluate:

• Take values of \(x\) slightly less than \(\frac{\pi}{4}\).
• Calculate corresponding \(\sin x\) values.
• These values approximate \(\lim_{x \to \frac{\pi}{4}^-} f(x) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\).

This calculation reveals how the function behaves from the left side, contributing to the understanding of the overall limit at the point.

Right-Hand Limit

A right-hand limit looks at the behavior as the input approaches a certain point from the right. This is expressed as \(\lim_{x \to c^+} f(x)\). In our exercise, focusing on \(x = \frac{\pi}{4}\), we look at values slightly more than \(\frac{\pi}{4}\) with \(f(x) = \cos x\).Steps to determine this limit include:

• Select values of \(x\) just greater than \(\frac{\pi}{4}\).
• Compute the \(\cos x\) for these points.
• The data approximates the limit \(\lim_{x \to \frac{\pi}{4}^+} f(x) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\).

This examination highlights how the function transitions to the right of \(\frac{\pi}{4}\), completing the picture necessary for interpreting the continuity of the function. Understanding both left and right-hand limits aids in discerning any discontinuities or peculiarities at specific points.