L'Hopital's Rule — 0/0 and ∞/∞ forms
Limits & Continuity
9
JEE Qs
8%
Hard
60
min
Master differentiation thoroughly and always check the indeterminate form before applying L'Hopital's Rule, simplifying after each step.
🧮 Key Formulas
✅ Key Points for JEE
- 1L'Hopital's Rule is applicable ONLY for indeterminate forms 0/0 and ∞/∞. Other indeterminate forms (0*∞, ∞-∞, 1^∞, 0^0, ∞^0) must be converted into one of these two forms first.
- 2When applying the rule, differentiate the numerator and the denominator SEPARATELY with respect to the variable, do NOT use the quotient rule for differentiation.
- 3The rule can be applied repeatedly as long as the limit continues to result in an indeterminate form (0/0 or ∞/∞) after each differentiation.
- 4Always simplify the resulting expression after each differentiation step before re-evaluating the limit to avoid unnecessary complexity in subsequent differentiations.
- 5L'Hopital's Rule is a powerful shortcut, but sometimes algebraic manipulation, factorization, or series expansion might be quicker or necessary if the derivatives become too complex.
⚠️ Common Mistakes
- ✕Applying L'Hopital's Rule when the limit is NOT of an indeterminate form (0/0 or ∞/∞), leading to incorrect results.
- ✕Incorrectly applying the quotient rule (d/dx [f(x)/g(x)] = [f'g - fg']/g²) instead of differentiating f(x) and g(x) separately.
- ✕Making errors in basic differentiation, especially with complex functions, trigonometric, exponential, or logarithmic derivatives.
- ✕Not simplifying the expression after differentiation, which can lead to more cumbersome calculations or further differentiation errors.
📝 Practice Questions
See allQ7. (2x2−3x+5)(3x−1) 2 limx→∞ is equal to : (3x2+5x+4)√(3x+2)x (1) 2 (2) 2e √3e √3 (3) 2 (4) 2e 3√e 3
Q19.Consider the region R = {(x, y) : x ≤y ≤9 −113 x2, x ≥0}. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R , is: (1) 730 (2) 625 119 111 (3) 821 (4) 567 123 121
Q11.If limx→∞(( 1−e ) ( e − 1+x )) = α, then the value of 1+loge α equals : (1) e−1 (2) e2 (3) e−2 (4) e
Q7. x2 {sin (k1 + 1)x + sin (k2 −1)x}, x < 0 ⎧ If the function f(x) = 4, x = 0 is continuous at x = 0, then k21 + k22 is ⎨ 2 2+k1x x > 0 x loge ( 2+k2x ), ⎩ equal to (1) 20 (2) 5 (3) 8 (4) 10
Q14. IfI(m, n) = ∫10 xm−1(1 −x)n−1dx, m, (1) I(19, 27) (2) I(9, 1) (3) I(1, 13) (4) I(9, 13)
Q9. Let [x] denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function f(x) = [x] + |x −2|, −2 < x < 3, is not continuous and not differentiable. Then m + n is equal to : (1) 6 (2) 8 (3) 9 (4) 7
NCERT Chapters
- Class 11 Maths Ch 13: Limits and Derivatives
- Class 12 Maths Ch 5: Continuity and Differentiability