Step 1: Understanding the Question:
We are given a piecewise function which is defined differently at and for all other values of . We are told the function is continuous at and we need to find the value of the constant that makes this true.
Step 2: Key Formula or Approach:
For a function to be continuous at a point , the following condition must be met:
In this problem, . We will also need the fundamental trigonometric limit:
Step 3: Detailed Explanation:
(i) Apply the condition for continuity at x=0:
For to be continuous at , we must have:
We are given that . So, the condition becomes:
(ii) Calculate the limit of f(x) as x approaches 0:
For the limit, we use the definition of when :
We can take the constant outside the limit:
Using the standard limit , we get:
(iii) Equate the limit and the function value:
From step (i) and (ii), we equate the two results:
and Therefore, for continuity:
Step 4: Final Answer:
The value of is 3.