Proving the Well-Known Limit of Sin(theta)/theta as Theta Approaches 0 = 1 Using Squeeze Theorem and L’Hopital’s Rule

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

1

The limit of sin(theta) / theta as theta approaches 0 is equal to 1. This is a well-known limit in calculus and can be proven using various methods such as L’Hopital’s Rule or the squeeze theorem.

One way to prove this is to use the squeeze theorem. We know that -1 ≤ sin(theta) ≤ 1 for all values of theta, so we can write:

-1 ≤ sin(theta) / theta ≤ 1/theta

As theta approaches 0, 1/theta approaches infinity, so we have an indeterminate form of 0/0. However, using the squeeze theorem, we can see that sin(theta) / theta is sandwiched between -1 and 1/theta, both of which approach 0 as theta approaches 0. Therefore, by the squeeze theorem, we can conclude that the limit of sin(theta) / theta as theta approaches 0 is 1.

Another way to prove this is to use L’Hopital’s Rule. Taking the derivative of both the numerator and denominator of sin(theta) / theta as theta approaches 0, we get:

lim theta->0 [cos(theta) / 1] = 1

Since the limit of the derivative is a finite number, we can conclude that the original limit of sin(theta) / theta as theta approaches 0 is also equal to 1.

Therefore, the correct answer is 1.

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