Problem 5 Determine the proper syntax for ... [FREE SOLUTION]

Chapter 1: Problem 5

Determine the proper syntax for evaluating \(\lim _{x \rightarrow \pi / 2} \sin x\) and evaluate the limit.

Short Answer

Expert verified

The limit is 1.

Step by step solution

Understand the limit notation

The limit notation \(\backslash lim _{x \rightarrow\ \backslash pi / 2}\backslash \text{sin} x\) means finding the value that the function \(\text{sin} x\) approaches as x approaches \(\backslash pi / 2\).

Substitute the value

In this specific case, direct substitution can be used because \(\text{sin} x\) is continuous at \(\backslash pi / 2\). Therefore, substitute \ x = \backslash pi / 2\ directly into the function.

Evaluate the sine function

Evaluate \(\backslash \text{sin} \left(\backslash \frac{\backslash\backslash pi\backslash}{2}\right) = 1\). This simplifies to \(\backslash sin \backslash pi/2 = 1\), because the sine of \(\backslash pi / 2\) is 1.

Conclusion

Therefore, the limit of \(\backslash sin x\) as \ x\ approaches \(\backslash \backslash pi / 2\) is 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit of a Function

Limits are fundamental to calculus and analysis. They help us understand the behavior of functions as they approach a certain point. In this exercise, we need to find the limit of the sine function as x approaches \(\frac{\pi}{2}\). This is written in limit notation as \(\text{lim}_{x \rightarrow \frac{\backslash \backslash pi}{2}} \text{sin} x\). The limit tells us what value the function \(\text{sin} x\) gets close to when x gets close to \(\frac{\backslash \backslash pi}{2}\).

Continuity

Continuity of a function at a point means that the function is well-behaved there, without any breaks or holes. For this exercise, the function \(\text{sin} x\) is continuous at \(\frac{\pi}{2}\). This means that as we get closer to \(\frac{\backslash \backslash pi}{2}\), the value of \(\text{sin} x\) gets closer to \(\text{sin} \frac{\backslash \backslash pi}{2}\). Because \(\text{sin} x\) is continuous at \(\frac{\backslash \backslash pi}{2}\), we can directly substitute \(\frac{\backslash \backslash pi}{2}\) into the function.

Trigonometric Functions

Trigonometric functions, like \(\text{sin} x\) and \(\text{cos} x\), describe relationships in a triangle and on the unit circle. They are periodic and oscillate between set values. For instance, \(\text{sin} x\) oscillates between -1 and 1. In our exercise, we are evaluating \(\text{sin} \frac{\backslash \backslash pi}{2} \). On the unit circle, \(\frac{\backslash \backslash pi}{2} \) radians corresponds to 90 degrees, where the sine value is 1. Hence, \(\text{sin} \frac{\backslash \backslash pi}{2} = 1\).

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91影视

Determine the proper syntax for evaluating \(\lim _{x \rightarrow \pi / 2} \sin x\) and evaluate the limit.

Short Answer

Step by step solution

Understand the limit notation

Substitute the value

Evaluate the sine function

Conclusion

Key Concepts

Limit of a Function

Continuity

Trigonometric Functions

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