Finding One-Sided Limits Algebraically


Find the limits in Exercises 11–20.


a. limx→0+ (1 − cos x) / |cos x − 1|

Verified step by step guidance

1. Identify the expression for the one-sided limit: \( \lim_{x \to 0^+} \frac{1 - \cos x}{|\cos x - 1|} \).
2. Recognize that as \( x \to 0^+ \), \( \cos x \to 1 \). Therefore, the expression \( \cos x - 1 \) approaches 0.
3. Since we are dealing with a one-sided limit from the right (\( x \to 0^+ \)), consider the behavior of \( \cos x - 1 \). For values of \( x \) slightly greater than 0, \( \cos x - 1 \) is negative, making \( |\cos x - 1| = -(\cos x - 1) \).
4. Substitute \( |\cos x - 1| \) with \( -(\cos x - 1) \) in the limit expression: \( \lim_{x \to 0^+} \frac{1 - \cos x}{-(\cos x - 1)} \).
5. Simplify the expression: \( \lim_{x \to 0^+} \frac{1 - \cos x}{-(\cos x - 1)} = \lim_{x \to 0^+} -1 \). Evaluate the limit as \( x \to 0^+ \).

Key Concepts

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side, either the left (denoted as lim x→c-) or the right (denoted as lim x→c+). Understanding one-sided limits is crucial for analyzing functions that may behave differently when approaching a point from different directions.

Absolute Value Function

The absolute value function, denoted as |x|, measures the distance of a number x from zero on the number line, always yielding a non-negative result. In the context of limits, the behavior of the absolute value can affect the limit's value, especially when the expression inside the absolute value changes sign around the point of interest.

Trigonometric Limits

Trigonometric limits involve evaluating the limits of functions that include trigonometric expressions, such as sine and cosine. These limits often require knowledge of trigonometric identities and properties, particularly as they relate to continuity and the behavior of these functions near specific points, such as zero.