Exploring the Limit as x Approaches 0 of sin(x): Understanding Sine Approximation and Formal Definition of a Limit

lim x->0 sinx =

The limit as x approaches 0 of sin(x) is 0

The limit as x approaches 0 of sin(x) is 0.

To see why, we can use the trigonometric identity that relates sin(θ) to θ for small angles. This identity is often referred to as the Small Angle Approximation or Sine Approximation.

The identity states that for small angles, sin(θ) is approximately equal to θ. In other words, when θ is close to 0, sin(θ) is close to θ.

So, as x approaches 0, sin(x) approaches x. Therefore, the limit as x approaches 0 of sin(x) is 0.

We can also confirm this limit using the formal definition of a limit. According to the formal definition, for any positive number ε, there exists a positive number δ such that if 0 < |x - 0| < δ, then |sin(x) - 0| < ε. To prove this, we can use the fact that the sine function is bounded between -1 and 1. This means that for any value of x, |sin(x)| ≤ 1. Let's choose δ = ε. If 0 < |x - 0| < δ, then |x| < ε. In this case, we have |sin(x) - 0| = |sin(x)| ≤ 1. Since |sin(x) - 0| = |sin(x)| ≤ 1 for all values of x, we can conclude that for any positive number ε, there exists a positive number δ (specifically δ = ε) such that if 0 < |x - 0| < δ, then |sin(x) - 0| < ε. This satisfies the formal definition of a limit. Therefore, we can confidently state that the limit as x approaches 0 of sin(x) is indeed 0.

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