Problem 56 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(55 - 58 ,\) complete parts \(( a ) - ( d )\) for the piecewise- definedfunction. \(\quad (\) a) Draw the graph of \(f\) . (b) At what points \(c\) in the domain of \(f\) does \(\lim _ { x \rightarrow c } f ( x )\) exist? (c) At what points \(c\) does only the left-hand limit exist? (d) At what points \(c\) does only the right-hand limit exist? $$f ( x ) = \left\\{ \begin{array} { l l } { \cos x , } & { - \pi \leq x < 0 } \\\ { \sec x , } & { 0 \leq x \leq \pi } \end{array} \right.$$

Step-by-Step Solution

Answer

(a) The graph of \(f\) is a cosine function from \(-\pi\) to 0 and a secant function from 0 to \(\pi\), with a hole at \(x=0\). (b) For all points \(c\) except \(c=0\), \(\lim _ { x \rightarrow c } f(x)\) exists. (c) At \(x=0\), only the left-hand limit exists. (d) At \(x=0\), only the right-hand limit exists.

Step 1: Sketching the Graph

First, you need to graph the piecewise function. In the interval -\(\pi \leq x < 0\) , \(f(x)=\cos x\) is a trigonometric function that oscillates between 1 and -1. For \(0 \leq x \leq \pi\), \(f(x) = \sec x\) is the reciprocal of \(\cos x\) which has vertical asymptotes at \(x = n\pi/2\), where n is an odd integer, and is undefined whenever \(\cos x = 0\). At \(x = 0\), there is a discontinuity.

Step 2: Identifying where the limit exists

Here, the function \(f(x)\) is continuous everywhere except at \(x = 0\). Therefore, at all points \(c\) in the domain, except for \(x = 0\), the limit \(\lim_{x \rightarrow c} f(x)\) exists.

Step 3: Identifying where only the left-hand limit exists

The left-hand limit exists at points \(c\) where the function is defined immediately to the left of \(c\), but not at \(c\). Our function \(f(x)\) is defined everywhere to the left of \(x = 0\), but not at \(x = 0\), thus the left-hand limit exists at \(x = 0\).

Step 4: Identifying where only the right-hand limit exists

The right-hand limit exists at points \(c\) where the function is defined immediately to the right of \(c\), but not at \(c\). Since \(f(x)\) is defined everywhere to the right of \(x = 0\) but not at \(x = 0\), the right-hand limit exists at \(x = 0\).

Key Concepts

Understanding Limits in Piecewise Functions

In mathematics, the concept of limits is fundamental to calculus and understanding piecewise functions. A limit attempts to find out what value a function gets closer to as the input approaches a particular point, say \(c\). For piecewise functions, like the one in the exercise, the behavior can be different on either side of that point because the function might be defined by different expressions.

• If both the left-hand and right-hand limits at \(c\) are equal, then the limit \(\lim_{x \rightarrow c} f(x)\) exists.
• If they are not equal, the overall limit does not exist at that point.

In the exercise, \(\lim_{x \rightarrow c} f(x)\) exists everywhere in the domain except at \(x = 0\) due to a switch from \(\cos x\) to \(\sec x\). This is a classic case where the left-hand and right-hand limits differ.

The Role of Continuity in Calculus

Continuity is when a function has no breaks, jumps, or holes at a point \(c\). For a function to be continuous at a point, the following must hold: \(f(c)\) is defined, \(\lim_{x \rightarrow c} f(x)\) exists, and they must be equal. In piecewise functions, continuity often breaks where one piece ends and another begins.

In the example provided, \(f(x)\) is not continuous at \(x = 0\) because there's a change from \(\cos x\) to \(\sec x\). At this point, though both functions might be defined individually, they do not connect smoothly. Recognizing where a function is or isn't continuous is crucial to solving calculus problems effectively.

Identifying Discontinuities in Functions

Discontinuities are points where a function is not continuous. In a piecewise function, these occur at the borders between different pieces. Detecting discontinuities is vital because they indicate where the function can't be graphically represented smoothly or doesn't have a limit.

Types of discontinuities include:

• Jump Discontinuity: The function jumps from one value to another as \(x\) approaches \(c\).
• Infinite Discontinuity: The function approaches infinity, often seen in asymptotes.
• Removable Discontinuity: A hole in the graph which could be "fixed" by redefining \(f(c)\).

In this exercise, there's a vertical asymptote introduced by \(\sec x\) at \(x = 0\), which is a type of infinite discontinuity.

Understanding Trigonometric Functions in Calculus

Trigonometric functions often appear in piecewise functions due to their cyclic nature. Each trigonometric function has unique characteristics that affect how limits and continuity are approached.

• Cosine Function (\(\cos x\)): This repetitive function oscillates between -1 and 1, making it well-behaved overall. It is continuous where it's defined, except at transitions within piecewise definitions.
• Secant Function (\(\sec x\)): This is the reciprocal of cosine, \(\sec x = 1/\cos x\), and it's undefined where \(\cos x = 0\). It has vertical asymptotes where it approaches infinity.

In the problem, \(f(x)\) switches from \(\cos x\) to \(\sec x\) at \(x = 0\). Understanding how these two functions behave and change as \(x\) approaches different values is key when tackling such exercises.