Evaluate lim x→0 x + 1/ 1 −cos x.

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Step 1: Recognize that the limit involves an indeterminate form. As x approaches 0, both the numerator and the denominator approach 0, which is an indeterminate form of type 0/0.

Step 2: Apply L'Hôpital's Rule, which is used to evaluate limits of indeterminate forms. According to L'Hôpital's Rule, if the limit of f(x)/g(x) as x approaches a value results in 0/0 or ∞/∞, then the limit can be found by differentiating the numerator and the denominator separately.

Step 3: Differentiate the numerator and the denominator. The numerator is x + 1, and its derivative is 1. The denominator is 1 - cos(x), and its derivative is sin(x).

Step 4: Rewrite the limit using the derivatives: lim x→0 (1/sin(x)).

Step 5: Evaluate the new limit. As x approaches 0, sin(x) approaches 0, so the limit becomes 1/0, which indicates the limit does not exist or is infinite.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of the expression as x approaches 0. Understanding limits is crucial for evaluating functions that may not be directly computable at specific points, especially when dealing with indeterminate forms.

Indeterminate Forms

Indeterminate forms occur when the direct substitution of a limit results in an undefined expression, such as 0/0 or ∞/∞. In the given limit, substituting x = 0 leads to the form 0/0, which requires further analysis, often using techniques like L'Hôpital's Rule or algebraic manipulation to resolve the limit and find a meaningful value.

Trigonometric Limits

Trigonometric limits involve the evaluation of limits that include trigonometric functions, such as sine and cosine. In this problem, the expression includes cos(x), and understanding the behavior of cosine near 0 is essential. Notably, the limit of (1 - cos(x)) as x approaches 0 can be simplified using trigonometric identities or Taylor series expansion, which is key to solving the limit.

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