$\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1$ where $\theta$ is measured in
A. radian
B. degree
C. grade
D. None of them
Answer (2)
The limit lim θ → 0 θ s i n θ = 1 is valid only when θ is measured in radians. If θ is in degrees, the limit does not hold true. Therefore, the correct answer is A. radian .
;
The limit lim θ → 0 θ s i n θ = 1 is given.
The proof of this limit relies on geometric arguments where θ is measured in radians.
If θ is measured in degrees, the limit would not equal 1.
Therefore, θ must be measured in radian .
Explanation
Problem Analysis The question states that lim θ → 0 θ s i n θ = 1 and asks in which units θ must be measured for this limit to hold true. We need to determine whether the angle θ should be in radians, degrees, grades, or none of these.
Understanding the Limit's Proof The standard proof of the limit lim θ → 0 θ s i n θ = 1 relies on geometric arguments and the sandwich theorem. Specifically, it involves comparing the area of a sector of a circle with the areas of triangles inscribed and circumscribed about the sector. The area of a sector with radius r and angle θ (in radians) is given by 2 1 r 2 θ . If θ were measured in degrees, the area of the sector would be 360 π r 2 θ .
Determining the Correct Unit The limit lim θ → 0 θ s i n θ = 1 is fundamentally defined and proven using radians. When using degrees, the limit becomes lim θ → 0 θ s i n ( 180 π θ ) = 180 π . Therefore, the original limit holds true only when θ is measured in radians.
Final Answer Therefore, the correct answer is that θ must be measured in radians for the limit to be equal to 1.
Examples
In physics, when analyzing simple harmonic motion, the angular frequency is often expressed in radians per second. The small angle approximation, where sin ( θ ) ≈ θ , is valid only when θ is measured in radians. This approximation simplifies many calculations in mechanics and wave phenomena, making radians the natural unit for angular measurements.
