The Sine Theta Over Theta Limit | Evaluating the Fundamental Limit in Calculus using Trigonometric Properties and the Squeeze Theorem

lim theta->0 sin(theta) / theta = ___________________

The limit you asked for is a fundamental limit in calculus and it is commonly referred to as the “sine theta over theta” limit

The limit you asked for is a fundamental limit in calculus and it is commonly referred to as the “sine theta over theta” limit. To evaluate this limit, we can use a basic trigonometric property and a concept known as the squeeze theorem.

Let’s start by using a well-known trigonometric property: as theta approaches zero, we have sin(theta) / theta = 1. This property comes from the geometric interpretation of sine as the ratio of the opposite side to the hypotenuse in a right-angled triangle, and for very small theta, the ratio essentially becomes 1.

To prove this property, we can use the squeeze theorem. The squeeze theorem states that if we have three functions, f(x), g(x), and h(x), such that f(x) ≤ g(x) ≤ h(x) for all x in some interval except possibly at x = a, and if both f(x) and h(x) approach the same limit L as x approaches a, then g(x) also approaches L as x approaches a.

In our case, let f(theta) = sin(theta) and h(theta) = theta. It is clear that f(theta) ≤ g(theta) ≤ h(theta) for all theta. Now, as theta approaches 0, we know that sin(theta) approaches 0, and theta obviously also approaches 0. Therefore, we have:

0 ≤ sin(theta) / theta ≤ 1

Now, taking the limit as theta approaches 0 for all three sides of the inequality, we conclude that:

lim theta->0 0 ≤ lim theta->0 (sin(theta) / theta) ≤ lim theta->0 1

Thus, we have:

0 ≤ lim theta->0 (sin(theta) / theta) ≤ 1

Since both 0 and 1 are the same limits in this case, we can safely say that:

lim theta->0 (sin(theta) / theta) = 1

Therefore, the limit you asked for is equal to 1.

Note: This is a proof of the limit, which may be beyond the scope of a typical high school math class. However, in practice, this limit is often accepted without being proven every time, as it is well-known and established.

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