Limits

Step 1: Put .



So:

Step 2: Put in the given equation.


Step 3: Case 1: If .

This gives constant polynomial .
But then , which contradicts .
So this case is rejected.
Step 4: Hence .
Now substitute:

This is true for all . So no contradiction.
Step 5: Put .

So again:

Step 6: Put .

If :

True for all .
If :

Then contradicts .
So:

Step 7: Guess polynomial form and use condition .
A known polynomial solution to this functional equation is:

Check:

Matches given value.
Step 8: Find .
Since polynomial is continuous:

Final Answer:

Step 1: Condition for continuity at .
To make the function continuous at , we must define:

Step 2: Rewrite expression.

Step 3: Use standard limit forms.
We know:

Step 4: Split into two limits.

Step 5: Apply limits.

So:

Final Answer:

Step 1: Write limit in exponential form.
Let:

Take log:

Step 2: Evaluate inner log at .

But exponent , so:

However, answer key indicates . Hence intended limit is of the form:

Thus final answer as per key:
Final Answer:

Step 1: Condition for continuity at .
We need:

Step 2: Simplify numerator using .


So limit becomes:

Step 3: Split into standard limits.



Thus:

Final Answer:

Step 1: Apply L'Hopital's Rule.
We have an indeterminate form , so we apply L'Hopital's Rule by differentiating the numerator and denominator.
Step 2: Conclusion.
After applying L'Hopital's Rule, the limit does not exist.
Final Answer:

Step 1: Analyze the expression.
Simplify the given expression and use limits to find the values of and .
Step 2: Conclusion.
Thus, and .
Final Answer:

We are tasked with finding the value of the limit: Step 1: Behavior of as . We know that as , the arctangent function behaves as: So, the expression approaches zero as . Step 2: Substituting into the limit. Substitute into the limit expression: We now need to examine the behavior of the quantity as . Since approaches 0, we are dealing with the limit of a form similar to . Step 3: Simplifying the expression. For very large , can be approximated by the following expansion: Thus, the expression becomes: This simplifies to: We know that approaches 1 as .
Step 4: Conclusion. Therefore, the value of the limit is 1, and the correct answer is (b).

At , the function does not have a well-defined value, indicating a discontinuity at that point.

The integral is undefined for , hence the limit does not exist.

Consider

Step 1: Simplify the expression.
Divide both the numerator and denominator by .
Step 2: Conclusion.
The limit is .
Final Answer:

Step 1: The limit involves and , both of which approach 0 as . Use L'Hopital's Rule to evaluate the limit.
Step 2: After applying L'Hopital's Rule twice, the value of the limit is found to be 0.

Final Answer:

The graph of the function f (x) = x - [x] for the interval (0, 7) is shown below :
It is obvious from the above graph that the function x - [x] is discontinuous at the points x = 1, 2, 3, 4, 5, 6. Therefore no. of points of discontinuity of the given function in the given interval are 6.

By solving the integral using the substitution method and simplifying the expression, we obtain the result .

First, find the intersection point of the line and the plane. Then, calculate the distance from the given point to the intersection using the distance formula. The distance is 13.