To solve the limit as x approaches 1 for the expression √(1 - cos²(x - 1)) / (x - 1), we need to analyze the behavior of the function as x gets very close to 1. Let's break this down step-by-step.
Understanding the Function
First, we can simplify the expression inside the limit. Notice that 1 - cos²(θ) can be rewritten using the identity for sine, since 1 - cos²(θ) = sin²(θ). Thus, our limit becomes:
- Limit as x → 1 of √(sin²(x - 1)) / (x - 1)
Applying the Absolute Value
Since the square root of a square gives the absolute value, we can express this as:
- Limit as x → 1 of |sin(x - 1)| / (x - 1)
Using L'Hôpital's Rule
As x approaches 1, both the numerator |sin(x - 1)| and the denominator (x - 1) approach 0, which presents a 0/0 indeterminate form. We can apply L'Hôpital's Rule, which states that if the limit results in an indeterminate form, we can take the derivative of the numerator and the denominator:
- Numerator: The derivative of |sin(x - 1)| is cos(x - 1) (considering the positive branch, since x is approaching 1 from either side).
- Denominator: The derivative of (x - 1) is simply 1.
Now, we can rewrite the limit:
- Limit as x → 1 of cos(x - 1) / 1
Evaluating the Limit
As x approaches 1, (x - 1) approaches 0. Therefore, cos(x - 1) approaches cos(0), which equals 1. This gives us:
Final Step: Putting It All Together
Thus, the limit exists and equals 1. This means that the options provided do not include the correct answer. To summarize:
- The limit exists and is equal to 1, which is not among the choices A, B, C, or D.
In conclusion, we’ve determined that the limit of the given expression as x approaches 1 is indeed 1, demonstrating that the function behaves nicely at that point. If you have any further questions about limits or need clarification on any steps, feel free to ask!
