Limit of the Cosine Function cos(1/x) as x Approaches Infinity: A Step-by-Step Guide
Step 1: Identify the Function
Math: lim (x → ∞) cos(1/x)
Explanation: We start by identifying the limit of the cosine function cos(1/x) as x approaches infinity, a fundamental concept in calculus.
Step 2: Analyze the Continuity of the Cosine Function
Explanation: The cosine function is continuous for all real numbers, allowing us to slip the limit inside the function. This is key in finding the limit of the cosine function cos(1/x) as x approaches infinity.
Step 3: Analyze the Argument of the Cosine Function
Math: lim (x → ∞) 1/x = 0
Explanation: As x approaches infinity, the value of 1/x approaches 0. This observation is crucial in understanding the limit of the cosine function cos(1/x) as x approaches infinity.
Step 4: Apply the Limit to the Cosine Function
Math: lim (x → ∞) cos(1/x) = cos(lim (x → ∞) 1/x) = cos(0)
Explanation: Since the cosine function is continuous for all real numbers, we can slip the limit inside, leading to cos(0). This step is essential in the limit of the cosine function cos(1/x) as x approaches infinity.
Step 5: Evaluate the Cosine of 0 Using the Unit Circle
Math: cos(0) = 1
Explanation: Using the unit circle, we know that the cosine of 0 is 1, providing the answer to the limit of the cosine function cos(1/x) as x approaches infinity.
Conclusion
The limit of the cosine function cos(1/x) as x approaches infinity is 1. This step-by-step guide, focusing on continuity for all real numbers, the process of slipping the limit inside the function, and the unit circle, offers valuable insights for students and educators alike. Explore the fascinating world of limits with this guide to the limit of the cosine function cos(1/x) as x approaches infinity.
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