# Limit & Continuity Important Questions-IIT JEE

IIT JEE Mathematics by -Jitendra Singh

To determine: The left and right hand limit of the function at given points and to check the continuity from left or right.

Solution:  left and right limits exist at x=0 but do not exist at x=2,4 and also the function is continuous from left and right at x=0.

Explanation: The left hand limit of function at a point is the value of function at which the function is approaching from the left and the right hand limit, the function approaching from right. Further if left and right hand limit are same then the limit of function at that point is exist.

Given:

The graph of function is given

Calculation:

As we seen from graph the function value is approaching to 1 when x is approaching from right as well from left. This means the left and right hand limit at x=0  is 1 so the limit of function at x=0 is 1.

At x=2, when x approaches to 2 from left or right then function is not approaching to finite value so the left and right hand limit does not exist.

At x=4, when x approaches to 4 from left or right then function is not approaching to finite value so the left and right hand limit does not exist.

Conclusion: The left and right hand limit of function is exist at x=0 but do not exist at x=2,4. And also the function is continuous from left and right at x=0 but does not at x=2,4.

To determine: The graph of the function with given conditions.

Solution: The graph is the answer that described in calculation section.

Explanation: The left hand limit of function at a point is the value of function at which the function is approaching from the left and the right hand limit, the function approaching from right. Further if left and right hand limit are same then the limit of function at that point is exist.

Given:

The left and right hand limits of function at x=2 are given as

To determine: The graph of the function and describe the left or right continuity at x=0.

Solution: The graph is the answer that described in calculation section and function is left-continuous at x=0 but not right-continuous.

Explanation: The left hand limit of function at a point is the value of function at which the function is approaching from the left and the right hand limit, the function approaching from right. Further if left and right hand limit are same then the limit of function at that point is exist.

Given:

To determine: The graph of the function by using given conditions.

Solution: The graph is the answer that described in calculation section.

Explanation: The left hand limit of function at a point is the value of function at which the function is approaching from the left and the right hand limit, the function approaching from right. Further if left and right hand limit are same then the limit of function at that point is exist.

Given:

The left hand limit of function at x=-3 is not defined and also the right hand limit at x=-3 is not defined as well as the limit at  x=4 is not exist i.e.

To determine: The point of discontinuities by using the two sided limits of given function and also the type of discontinuity.

Solution: The function is discontinuous at x=-1 and type of discontinuity is removable.

Explanation: The left hand limit of function at a point is the value of function at which the function is approaching from the left and the right hand limit, the function approaching from right. Further if left and right hand limit are same then the limit of function at that point is exist.

Conclusion: The function is discontinuous at x= -1 because left and right – limit are distinct. The type of discontinuity is removable discontinuity.

To prove: The given function is continuous at each point of domain of the function.

Solution: The function is continuous at each point of domain.

Explanation: The left hand limit of function at a point is the value of function at which the function is approaching from the left and the right hand limit, the function approaching from right. Further if left and right hand limit are same then the limit of function at that point is exist.

If the limit of function is equal to the value of function at the point then function is continuous at that point.

Given:

Conclusion: The function is continuous at x = a  R. Hence function is continuous at domain of function.

To Determine: The constant b so that the function is continuous at x=2 .

Solution: The value of b is 7 and function is discontinuous at x=-2.

Explanation: The left hand limit of function at a point is the value of function at which the function is approaching from the left and the right hand limit, the function approaching from right. Further if left and right hand limit are same then the limit of function at that point is exist.

If the limit of function is equal to the value of function at the point then function is continuous at that point.