Continuity

Concept: A function is continuous at a point if the following condition holds: In this problem, the function must be continuous at . Therefore, We also use the important trigonometric limit:

Step 1: Apply the continuity condition. Given For continuity at :

Step 2: Evaluate the limit. Using the standard limit: Thus,

Step 3: Use the given value of the function. Since ,

Concept: A function is continuous at a point if: If the function gives an indeterminate form at that point, we evaluate the limit using standard techniques like L'Hôpital's Rule.
Step 1: {Substitute into the function.}
This is an indeterminate form.
Step 2: {Apply L'Hôpital's Rule.}

Step 3: {Evaluate the limit.}

Step 4: {For continuity, limit must equal function value.}
Since the function approaches , it must be finite and well-defined. Hence,