Pregunta
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Responder
The limit is equal to cos x.
Solución
To evaluate the limit:
This limit represents the derivative of the sine function at the point
. Let’s compute it step by step.
Step 1: Expand
using the sine addition formula
Step 2: Substitute the expansion back into the limit
Step 3: Split the limit into two parts
Step 4: Evaluate each limit separately
Step 5: Combine the results
Conclusion:
Answer:
The limit is equal to cos x.
The limit is equal to cos x.
Respondido por UpStudy AI y revisado por un tutor profesional
Como
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The Deep Dive
The limit you’re looking for is actually a fundamental concept in calculus, specifically in finding the derivative of the sine function. As
approaches 0, this expression represents the derivative of
at the point
. Using the definition of the derivative and applying L’Hôpital’s rule or recognizing the limit directly, we find that:
Now, isn’t it fascinating how this simple limit leads us to the derivative, thereby linking it to the broader field of trigonometric calculus? The sine function’s derivative being the cosine function is a beautiful example of how interconnected functions are in mathematics!
Additionally, understanding these limits is crucial in real-world applications such as physics and engineering, where they are used to analyze wave functions, oscillations, and other periodic phenomena. When studying these concepts, engineers often need to calculate velocities and accelerations which involve differentiating sine and cosine functions. Isn’t it amazing how mathematical theories translate into practical solutions?
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