Limit And Continuity

Step 1: Continuity condition
For continuity at :
Step 2: Find
Using middle function:
Step 3: Left-Hand Limit (LHL)
Equate with :
Step 4: Right-Hand Limit (RHL)
Equate with : Final Answer:

Step 1: Understanding the Concept:
The problem requires evaluating the right-hand limit (RHL) and the left-hand limit (LHL) of a piecewise-defined function at the point where its definition changes ( ), and then summing the two limit values.

Step 2: Key Formula or Approach:
- To find (RHL), use the piece of the function defined for . - To find (LHL), use the piece of the function defined for . - Substitute into these respective pieces to calculate the limits, then add them.

Step 3: Detailed Explanation:
1. Evaluate the Right-Hand Limit (RHL):
We are approaching 2 from values greater than 2 ( ).
For , the function is defined as .
Therefore, .
Substituting directly gives:
.
2. Evaluate the Left-Hand Limit (LHL):
We are approaching 2 from values less than 2 ( ).
For , the function is defined as .
Therefore, .
Substituting directly gives:
.
3. Calculate the sum:
The question asks for the sum of these two limits:
Sum
Sum .

Step 4: Final Answer:
The sum of the limits is 6.

Step 1: Understanding the Concept:
We are given a limit of a rational function that evaluates to a finite non-zero value ( ) as approaches . Since the denominator evaluates to zero ( ) at , the limit can only exist if the numerator also evaluates to zero at . This creates an indeterminate form of type .

Step 2: Key Formula or Approach:
1. Set the numerator expression to zero and substitute : . 2. Simplify the resulting equation to find a relationship between and .

Step 3: Detailed Explanation:
Let the given limit be . When we substitute directly into the denominator, we get . For the limit of the fraction to exist and be a finite number (like 5), the numerator must also approach zero as approaches 3. This avoids a situation, which would lead to an undefined limit ( ). Therefore, we must have: Substituting into the numerator: Now, simplify this equation to find the value of : Divide the entire equation by 3: Rearranging the terms yields:
Step 4: Final Answer:
The value of is 3.