Important Questions on Limits, Continuity, and Differentiability
FAQs on Limit Continuity and Differentiability Practice Paper
1. What is the meaning of continuity of a function in calculus?
A function is said to be continuous at a point if its value at that point matches its limit from both sides. This means:
- The function is defined at that point.
- The limit of the function exists as x approaches that point.
- The limit value equals the actual function value at that point.
2. How is the differentiability of a function defined?
A function is differentiable at a point if its derivative exists at that point. In simple terms:
- The function must be continuous at that point.
- The left-hand derivative and right-hand derivative must both exist and be equal.
3. What is the relationship between continuity and differentiability?
Every function that is differentiable at a point is also continuous at that point, but the reverse is not true. Key points include:
- Differentiability ⇒ Continuity.
- Continuity does not always guarantee differentiability (e.g., the function |x| at x=0 is continuous but not differentiable).
4. What is the limit of a function and how is it calculated?
The limit of a function at a point describes the value the function approaches as the input approaches that point. To calculate limits:
- Direct substitution if function is defined and continuous.
- Factorization or rationalization if direct substitution gives 0/0.
- Apply L'Hospital's Rule for indeterminate forms.
5. Explain the conditions required for a function to be continuous at a point x = a.
For a function f(x) to be continuous at x = a, these three conditions must be met:
- f(a) is defined
- limx→a f(x) exists
- limx→a f(x) = f(a)
6. What is the difference between left-hand limit and right-hand limit?
The left-hand limit (LHL) approaches a point from values less than the point, while the right-hand limit (RHL) approaches from greater values. For the limit to exist, both must be equal:
- LHL: limx→a− f(x)
- RHL: limx→a+ f(x)
7. Give an example of a function that is continuous but not differentiable.
A classic example is the absolute value function, f(x) = |x|. It is:
- Continuous at x = 0 (no break).
- Not differentiable at x = 0 (sharp corner).
8. What steps are involved in checking the differentiability of a function at a point?
To test for differentiability at x = a, follow these steps:
- Ensure the function is continuous at x = a.
- Compute the left-hand derivative at x = a.
- Compute the right-hand derivative at x = a.
- Check if both derivatives exist and are equal.
9. What are the standard limits often used in CBSE exams?
Some standard limits important for exams include:
- limx→0 (sinx)/x = 1
- limx→0 (1 - cosx)/x = 0
- limx→0 (ex - 1)/x = 1
10. How do you prove a function is not continuous at a given point?
To prove non-continuity at x = a, show that at least one of the conditions of continuity fails:
- Function not defined at x = a
- Limit does not exist as x → a
- Limit exists but does not equal f(a)
11. State the conditions for a function to be differentiable at a point.
A function f(x) is differentiable at x = a if:
- It is continuous at x = a
- Both left-hand and right-hand derivatives exist and are equal
12. Can a discontinuous function be differentiable? Explain.
No, a discontinuous function cannot be differentiable at the point of discontinuity. Differentiability always implies continuity, but the reverse is not true. Thus, if the function is not continuous at a point, its derivative does not exist there.
