MathsHardClass 12

Limits — Standard Forms + L'Hopital

Limits & Continuity

JEE Qs

20%

Hard

min

Prioritize algebraic simplification and standard limit forms; use L'Hopital's Rule strategically for 0/0 or ∞/∞ forms, and leverage series expansions for complex cases.

🧮 Key Formulas

lim (x->0) sin x / x = 1

lim (x->0) tan x / x = 1

lim (x->0) (1 - cos x) / x^2 = 1/2

lim (x->0) (e^x - 1) / x = 1

lim (x->0) (a^x - 1) / x = ln a

lim (x->0) ln(1 + x) / x = 1

lim (x->0) (1 + x)^(1/x) = e

lim (x->inf) (1 + a/x)^x = e^a

lim (x->inf) (1 + 1/x)^x = e

L'Hopital's Rule: If lim (x->a) f(x)/g(x) results in 0/0 or inf/inf, then lim (x->a) f(x)/g(x) = lim (x->a) f'(x)/g'(x)

If lim (x->a) f(x)^g(x) results in 1^inf, 0^0, or inf^0, use e^(lim (x->a) g(x)ln f(x))

✅ Key Points for JEE

1Master the recognition and transformation of all seven indeterminate forms: 0/0, ∞/∞, 0*∞, ∞-∞, 1^∞, 0^0, ∞^0.
2Prioritize standard limit forms (especially for trigonometric, exponential, and logarithmic functions) and algebraic manipulation, as they are often quicker and prevent calculus errors compared to L'Hopital's Rule.
3L'Hopital's Rule is strictly applicable only for 0/0 or ∞/∞ forms; differentiate the numerator and denominator *separately*, not as a quotient rule.
4For indeterminate forms like 1^∞, 0^0, and ∞^0, convert them using logarithms or the e^(lim f(x)ln g(x)) transformation before applying other limit techniques.
5Series expansions (Taylor/Maclaurin) of functions like sin x, cos x, e^x, ln(1+x), and (1+x)^n are powerful alternatives to L'Hopital's Rule, particularly for complex limits where multiple differentiations would be tedious.

⚠️ Common Mistakes

✕Applying L'Hopital's Rule when the limit is not in one of the required indeterminate forms (0/0 or ∞/∞).
✕Differentiating the entire fraction using the quotient rule instead of differentiating the numerator and denominator separately for L'Hopital's Rule.
✕Incorrectly handling or converting indeterminate forms like 0*∞ or ∞-∞ into 0/0 or ∞/∞, or incorrectly applying the exponential form for 1^∞, 0^0, ∞^0 limits.
✕Over-reliance on L'Hopital's Rule, overlooking simpler solutions available through standard limits or algebraic factorization/rationalization, leading to longer solution paths and increased error probability.

📝 Practice Questions

See all

Q7. (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

2025·MCQEasy

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

2025·MCQHard

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

2025·MCQHard

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

2025·Multi conceptHard

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)

2025·MCQHard

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

2025·MCQMedium

NCERT Chapters

Class 11 Maths Ch 13: Limits and Derivatives
Class 12 Maths Ch 5: Continuity and Differentiability