Evaluating The Limit Of (cos X - 1) / |cos X - 1| As X Approaches 0 From The Left

In the realm of calculus, evaluating limits is a fundamental concept. Limits help us understand the behavior of functions as their input approaches a specific value. In this article, we will delve into the limit of the function (cos x - 1) / |cos x - 1| as x approaches 0 from the left, denoted as $\lim _{x \rightarrow 0^{-}} \frac{\cos x-1}{|\cos x-1|}$. This particular limit involves the interplay between the cosine function, absolute values, and one-sided limits, making it an interesting case study. Understanding this limit requires a solid grasp of trigonometric functions, absolute values, and the concept of limits, particularly one-sided limits. We will explore the behavior of the cosine function near 0, how the absolute value function affects the expression, and ultimately, how these factors combine to determine the limit as x approaches 0 from the left.

To effectively evaluate the limit, we must first understand the behavior of the cosine function, cos x, near zero. The cosine function is a fundamental trigonometric function that oscillates between -1 and 1. At x = 0, cos(0) = 1. As x moves away from 0, the value of cos x decreases. For values of x near 0, the cosine function can be approximated using its Taylor series expansion, which is given by:

$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ...$

This series expansion reveals that for x close to 0, the dominant term is 1 - x²/2. Thus, cos x is slightly less than 1 for x near 0, whether x is positive or negative. This is a crucial observation because it determines the sign of the expression cos x - 1. Since cos x is slightly less than 1 near 0, cos x - 1 will always be negative in the vicinity of 0, except at x = 0 itself. Understanding this behavior is essential for dealing with the absolute value in our limit expression. The absolute value function, denoted by | |, returns the magnitude of a number, effectively making any negative number positive. Therefore, |cos x - 1| will always be the positive version of cos x - 1. This fact will play a critical role in simplifying our limit expression and arriving at the final answer. In the subsequent sections, we will delve deeper into how the absolute value function interacts with cos x - 1 and how this affects the overall limit.

The absolute value function, denoted as |x|, is a crucial component in the given limit problem. The absolute value of a number is its distance from zero, regardless of direction. Mathematically, it is defined as:

|x| = x, if x ≥ 0 |x| = -x, if x < 0

In our case, we have |cos x - 1|. As discussed earlier, cos x is slightly less than 1 for x near 0 (except at x = 0), which means that cos x - 1 is negative in the vicinity of 0. Therefore, according to the definition of the absolute value, |cos x - 1| will be equal to -(cos x - 1) when x is near 0 but not equal to 0. This transformation is vital because it allows us to simplify the expression inside the limit. Specifically, we can rewrite |cos x - 1| as 1 - cos x when cos x - 1 is negative, which is the case as x approaches 0 from either side (excluding x = 0). The absolute value function ensures that the denominator of our limit expression is always positive, which is essential for the limit to exist and be well-defined. Without the absolute value, we would encounter issues with the sign of the denominator, potentially leading to an incorrect limit evaluation. This understanding of the absolute value's role is key to simplifying the expression and accurately calculating the limit as x approaches 0 from the left. In the next section, we will see how this simplification allows us to evaluate the limit effectively.

Now that we understand the behavior of cos x near 0 and the effect of the absolute value function, we can evaluate the one-sided limit: $\lim _{x \rightarrow 0^{-}} \frac{\cos x-1}{|\cos x-1|}$.

As x approaches 0 from the left (denoted as x → 0⁻), x takes on negative values close to 0. As we’ve established, for x near 0 (but not equal to 0), cos x - 1 is negative. Therefore, |cos x - 1| = -(cos x - 1) = 1 - cos x.

Thus, we can rewrite the limit as:

$\lim _{x \rightarrow 0^{-}} \frac{\cos x-1}{|\cos x-1|} = \lim _{x \rightarrow 0^{-}} \frac{\cos x-1}{-( \cos x-1)}$

Since cos x - 1 is not equal to 0 as x approaches 0 from the left, we can cancel the terms in the numerator and the denominator:

$\lim _{x \rightarrow 0^{-}} \frac{\cos x-1}{-( \cos x-1)} = \lim _{x \rightarrow 0^{-}} -1$

The limit of a constant is simply the constant itself. Therefore,

$\lim _{x \rightarrow 0^{-}} -1 = -1$

This result shows that as x approaches 0 from the left, the function (cos x - 1) / |cos x - 1| approaches -1. The negative sign is a direct consequence of the absolute value function and the fact that cos x - 1 is negative near 0. This evaluation highlights the importance of understanding one-sided limits and the behavior of functions near specific points. In the following sections, we will discuss the implications of this result and compare it with the limit as x approaches 0 from the right, if it exists.

Having evaluated the limit $\lim _{x \rightarrow 0^{-}} \frac{\cos x-1}{|\cos x-1|} = -1$, we can now discuss the implications of this result. This one-sided limit tells us that as x approaches 0 from the left, the function (cos x - 1) / |cos x - 1| converges to -1. This behavior is due to the fact that for negative values of x close to 0, cos x is slightly less than 1, making cos x - 1 negative. The absolute value ensures that the denominator is always positive, leading to the negative sign in the limit.

To further understand the function's behavior, we should also consider the limit as x approaches 0 from the right, denoted as $\lim _{x \rightarrow 0^{+}} \frac{\cos x-1}{|\cos x-1|}$. For x approaching 0 from the right, cos x is still slightly less than 1, so cos x - 1 remains negative. Thus, |cos x - 1| is still equal to -(cos x - 1) = 1 - cos x. Therefore, the limit from the right is also:

$\lim _{x \rightarrow 0^{+}} \frac{\cos x-1}{|\cos x-1|} = \lim _{x \rightarrow 0^{+}} \frac{\cos x-1}{-( \cos x-1)} = \lim _{x \rightarrow 0^{+}} -1 = -1$

Since both the left-hand limit and the right-hand limit exist and are equal to -1, we can conclude that the two-sided limit also exists and is equal to -1:

$\lim _{x \rightarrow 0} \frac{\cos x-1}{|\cos x-1|} = -1$

In conclusion, by carefully analyzing the behavior of the cosine function near zero, understanding the role of the absolute value function, and evaluating the one-sided limits, we have determined that the limit of (cos x - 1) / |cos x - 1| as x approaches 0 is -1. This exercise illustrates the importance of a thorough understanding of these fundamental concepts in calculus and how they interact to determine the behavior of functions near specific points.

In summary, evaluating the limit $\lim _{x \rightarrow 0^{-}} \frac{\cos x-1}{|\cos x-1|}$ involves several key steps and concepts. First, we recognize the behavior of the cosine function near zero. As x approaches 0, cos x approaches 1, but for x near 0 (excluding 0), cos x is slightly less than 1. This means that cos x - 1 is negative in the vicinity of 0. Second, understanding the absolute value function is crucial. The absolute value of a number is its distance from zero, so |cos x - 1| is equal to -(cos x - 1) when cos x - 1 is negative. This allows us to simplify the expression by rewriting |cos x - 1| as 1 - cos x. Third, we evaluate the one-sided limit. As x approaches 0 from the left (x → 0⁻), we substitute |cos x - 1| with -(cos x - 1) in the limit expression. This simplifies the fraction, allowing us to cancel out the (cos x - 1) terms, resulting in a limit of -1. Fourth, we consider the implications of this result. Since the limit from the left is -1, we also check the limit from the right. In this case, the limit from the right is also -1, which means the two-sided limit exists and is -1. Finally, this problem highlights the importance of understanding the behavior of trigonometric functions, the role of absolute values, and the concept of one-sided limits in calculus. These concepts are fundamental for evaluating limits and understanding the behavior of functions near specific points. This comprehensive approach ensures an accurate and thorough evaluation of the limit.

Beyond the direct evaluation of the limit $\lim _{x \rightarrow 0^{-}} \frac{\cos x-1}{|\cos x-1|}$, there are several additional considerations and avenues for further exploration that can deepen our understanding of this topic. One important consideration is the graphical representation of the function f(x) = (cos x - 1) / |cos x - 1|. Plotting this function reveals a piecewise behavior. For x ≠ 0, f(x) is equal to -1, as we’ve shown through our limit evaluation. However, at x = 0, the function is undefined because the denominator becomes zero. This graphical perspective provides a visual confirmation of our analytical result, showing a constant value of -1 as x approaches 0 from either side. Another area for further exploration involves comparing this limit with similar limits involving other trigonometric functions or algebraic expressions. For instance, one could investigate the limit of (sin x) / |sin x| as x approaches 0 from different directions. This comparison can highlight the unique properties of each function and how they interact with the absolute value. Additionally, exploring the use of L'Hôpital's Rule in this context is worthwhile. While L'Hôpital's Rule is typically used for indeterminate forms like 0/0 or ∞/∞, it may not be directly applicable here due to the absolute value function. However, understanding why it cannot be applied and the alternative methods for evaluating such limits is an important learning experience. Furthermore, considering the broader applications of limits in calculus and other fields can provide valuable context. Limits are fundamental to the definition of continuity, derivatives, and integrals, and they play a crucial role in various areas of science and engineering. Exploring these applications can motivate a deeper understanding of limit concepts and their practical significance. In conclusion, while we have successfully evaluated the given limit, there are numerous avenues for further exploration and consideration that can enhance our understanding of calculus and its applications.