Functions

Step 1: Apply continuity condition.
For the function to be continuous on , the left-hand limit and right-hand limit at must be equal. Therefore, we need to equate the two expressions for at :
Step 2: Solve for .
Simplifying both sides, we get: Solving for , we get .
Step 3: Conclusion.
Thus, , making option (A) the correct answer.

Consider the function and analyze the transformations step by step: First, examine : Next, for : Finally, for : Thus, is not differentiable at the points . Final Answer:

Given two vectors, and . Step 1: Calculate Simplifying this expression: Step 2: Find the magnitude of The magnitude is given by: Simplifying further: Step 3: Calculate and For , we have: For , we compute: Step 4: Verify the relationship between the magnitudes
Now, we compare with : Since we know that , and , the inequality holds true: Final Answer:

We are given the line equation: Since point is equidistant from and , the condition must hold. Therefore, we use the equation: Step 1: Use the distance formula. The distances from to and are: Squaring both sides, we get: Simplifying: Combine like terms: Step 2: Solve the system of equations. From the line equation , we can express in terms of as: Substitute this into the simplified distance equation: Simplify: Solving for : Substitute into the equation for : Final Answer: The coordinates of point are:

Given the functions and and that , meaning: First compute both compositions: Then, equate the two expressions: Canceling out the terms: Solve for :

Step 1: Substitute into the function
We are given the function . We need to find . Substitute into the function:
Step 2: Simplify the expression

Answer:
Therefore, . So, the correct answer is option (2).