Limits, Continuity & Differentiability – JEE Main Practice Test 2026

Limits, Continuity & Differentiability – JEE Main Practice Test

This JEE Main Practice Test covers the most important and high-level questions from Limits, Continuity and Differentiability. All questions are strictly JEE Main oriented and designed to improve concept clarity and accuracy.

Q1. If lim (x→0) [ (αeˣ + βe⁻ˣ + γsinx) / (x sin²x) ] = 2/3, then which of the following is NOT correct?
α² + β² + γ² = 6
αβ + βγ + γα + 1 = 0
αβ² + βγ² + γα² + 3 = 0
α² − β² + γ² = 4
Q2. Suppose f : R → (0,∞) is differentiable and 5f(x+y) = f(x)f(y). If f(3) = 320, then ∑ f(n) from n=0 to 5 is:
6875
6525
6575
6825
Q3. If a = lim(x→0) [ (√(1+√(1+x⁴)) − √2) / x⁴ ] and b = lim(x→0) [ sin²x / (√2 − √(1+cosx)) ], then the value of ab³ is:
36
25
32
30
Q4. If f(x) = (sin3x + αsinx − βcos3x) / x³ is continuous at x = 0, then f(0) is equal to:
4
−2
−4
2
Q5. Let f(x) = (x² − 1)|x² − ax + 2| + cos|x| be not differentiable at x = α = 2 and x = β. Then distance of point (α,β) from line 12x + 5y + 10 = 0 is:
5
2
4
3
Q6. Let [x] be GIF. If f(x) = [x] + |x − 2|, −2 < x < 3, then m + n (points of non-continuity and non-differentiability) is:
6
9
8
7
Q7. Number of points of discontinuity of f(x) = [x²/2] − [√x], x ∈ [0,4], is:
Correct Answer: 8
Q8. Let m and n be number of points where f(x) = max{x, x³, x⁵, …, x²¹} is not differentiable and not continuous respectively. Then m + n is:
Correct Answer: 3
Q9. If f(x) = |[x]| + √(x − [x]), then number of points of discontinuity in (−2,1) is:
Correct Answer: 2
Q10. lim (x→0) [ e − (1+2x)^(1/2x) ] / x is equal to:
−2/e
e − e²
0
e
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