Limit And Continuity

Step 1: Understanding the Concept:
For a function to be continuous at a point , the left-hand limit (LHL), the right-hand limit (RHL), and the value of the function at that point, , must all be equal.
Step 2: Key Formula or Approach:
We will test the continuity at and by calculating the LHL, RHL, and the function's value at each point.
Step 3: Detailed Explanation or Calculation:
Testing continuity at :
1. Left-Hand Limit (LHL):
For , the function is .
2. Right-Hand Limit (RHL):
For , the function is .
3. Value of the function at :
For , the function is .
Since LHL = RHL = , the function is continuous at .
Testing continuity at :
1. Left-Hand Limit (LHL):
For , the function is .
2. Right-Hand Limit (RHL):
For , the function is .
3. Value of the function at :
For , the function is .
Since LHL = RHL = , the function is continuous at .
Step 4: Final Answer:
The function is continuous at both and .

Step 1: Understanding the Concept:
To show that the absolute value function is continuous everywhere, we can analyze it as a piecewise function and check for continuity at the point where its definition changes, which is at . For all other points, the function is a simple polynomial ( or ), which is known to be continuous.
Step 2: Key Formula or Approach:
The piecewise definition of is:
For continuity at a point , we must show .
Step 3: Detailed Explanation or Calculation:
Case 1:
For any , in a small neighborhood around c, . Also, . Since the limit equals the function value, is continuous for all .
Case 2:
For any , in a small neighborhood around c, . Also, . Since the limit equals the function value, is continuous for all .
Case 3:
We need to check the left-hand limit, right-hand limit, and the function value at .
- Left-Hand Limit (LHL): For , we use . - Right-Hand Limit (RHL): For , we use . - Function Value: .
Since LHL = RHL = , the function is continuous at .
Step 4: Final Answer:
Since the function is continuous for , , and at , it is continuous for all real values of x. Hence proved.

Step 1: Understanding the Concept:
A function is said to be continuous at a point if three conditions are met:
is defined.
The limit of the function as approaches exists. This means the Left-Hand Limit (LHL) equals the Right-Hand Limit (RHL). The value of the limit is equal to the value of the function at that point. Step 2: Detailed Explanation:
We need to check these three conditions for the function at the point .
Condition 1: Value of the function at x = 0
The value of the function at is: The function is defined at .
Condition 2: Existence of the limit at x = 0
We need to calculate the Left-Hand Limit (LHL) and the Right-Hand Limit (RHL).
Left-Hand Limit (LHL): By definition, for , . So, we can write: Alternatively, using : Right-Hand Limit (RHL): By definition, for , . So, we can write: Alternatively, using : Since LHL = RHL = 0, the limit exists and .
Condition 3: Comparing the limit and function value
We have found that: Therefore, .
Step 3: Final Answer:
Since all three conditions for continuity are satisfied, the function is continuous at .

Step 1: Understanding the Concept:
A function is continuous at a point if the following three conditions are met:
1. is defined.
2. The limit of as approaches exists. This means the left-hand limit (LHL) must equal the right-hand limit (RHL).
3. The limit equals the function value: .
Step 2: Key Formula or Approach:
We need to calculate the LHL, RHL, and and check if they are equal.
LHL:
RHL:
Step 3: Detailed Explanation or Calculation:
1. Calculate the Left-Hand Limit (LHL):
For , we use the definition because .
2. Calculate the Right-Hand Limit (RHL):
For , we use the definition because .
3. Compare the limits:
Here, LHL = 6 and RHL = -4.
Since , the limit does not exist.
Step 4: Final Answer:
Because the limit of the function as approaches 1 does not exist, the function is not continuous (it is discontinuous) at .