Integer Answer Type Questions for JEE Limits, Continuity & Differentiability
Q.1. If 
then find the value of a.
Ans. 1
Since, the given limit is in the form of 0/0, we will use L’ Hospital’s Rule
It is satisfied only when a = 1.
Q.2. 
Ans. 0
Q.3. 
Ans. 1
where α = x4 + x2 = 1
Q.4. The value of 
is
Ans. 3
Q.5. The value of 
must be
Ans. 8
Q.6. Let f(x) = [x]+ |1 - x|, -1 ≤ x ≤ 3 and [x] is the largest integer not exceeding x. The number of points in [-1, 3] where f is not continuous is
Ans. 4
f(x) = [x] + |1 - x|, -1 ≤ x ≤ 3
= -x if -1 ≤ x < 0
=1 - x if 0 ≤ x <1 = x if 1 ≤ x < 2
= x +1 if 2 ≤ x < 3 = 5 if x = 3
Clearly f is not continuous at x = 0, 1, 2 and 3
Q.7. Let f : R → R be a differential function satisfying 
for all real x and y and f'(2) = 2. If g(x) = f(| x |)- 3| for all x ∈ R , then for g(x) total non-differentiable point are
Ans. 3
⇒ f '(2) = f'(0) = 2 (∵ f'(2) = 2) ⇒ f'(x) = 2 ⇒f(x)= 2x + c
Put x = y = 0 in
Now, from equation (i), f(0) = 0 + c = 2
∴ c = 2
From equation (i), f(x) = 2x + 2
So, function g(x) = |2|x| - 1|, hence the points of non-differentiability of g(x) are
x = ±(1/2), 0
Q.8. Let f(x) = 
g(x) = f (x -1) + f (x + 1). Then the number of points where g is not differentiable is
Ans. 5
⇒ (x + y)2007 = xy ⇒ xy - 2007x - 2007y = 0
(x - 2007)(y - 2007) = 20072 = 34 x 2232
The number of pairs is equal to the number of divisors of 20072 that is (4 + 1) x (2 + 1) = 15
Since, x < y, so required number of pairs = 7
Q.9. Let f (x) = 
If m is the number of points in [0, 3]where f is discontinuous and n is the number of points in [0, 3]where f is not differentiable, then the value of m + n is
Ans. 5
f is discontinuous at x = 1 and x = 3 only
Also, f'(0) = 0 and f is not derivable at x = 2
So m = 2 and n = 3.
Q.10. Let f(x) = x tan-1(x2) + x4. Let fk(x) denotes kth derivative of f(x) w.r.t. x, k ∈ N. If f 2m(0) ≠ 0 m ∈ N then m equals
Ans. 2
Let g(x) = x tan-1 (x2). It is an odd function. So, g2m (0) = 0.
Let h(x) = x4.
So, f(x) = g(x) + h(x) ⇒ f2m (0) = g2m(0) + h2m(0) = h2m (0) ≠ 0.
It happens when 2m = 4 ⇒ m = 2.
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