JEE Maths›Limits, Continuity & Differentiability›Question Types

Limits, Continuity & Differentiability - How It Appears in JEE

8 recurring patterns. Learn the pattern, recognize it in 5 seconds, apply the right approach.

Evaluate Limit Using Standard Forms

Pattern

Direct application of standard limit results like sin x/x, (e^x - 1)/x, (1+x)^(1/x)

How to recognize

Expression involves trig/exponential/log functions near 0 or infinity, matches a known standard limit pattern

→ Identify which standard limit applies. Rewrite the expression to match the standard form exactly. Substitute and simplify. For composite forms like sin(3x)/(5x), split as (sin 3x)/(3x) * (3/5).

Evaluate Limit Using L'Hopital's Rule

Pattern

Limit results in 0/0 or infinity/infinity indeterminate form requiring differentiation

How to recognize

Direct substitution gives 0/0 or infinity/infinity. Standard forms don't apply directly or the expression is complex.

→ Verify the indeterminate form. Differentiate numerator and denominator separately. Re-check the form and apply again if needed. Convert other indeterminate forms (0 * infinity, infinity - infinity, 1^infinity) to 0/0 or infinity/infinity first.

Find Limit Using Taylor/Maclaurin Expansion

Pattern

Use series expansion to evaluate tricky limits, especially when L'Hopital is tedious

How to recognize

Limit involves differences like sin x - x, e^x - 1 - x, or ratios where multiple L'Hopital applications would be needed

→ Expand each function using Taylor series around the limit point. Keep terms up to the order needed for cancellation. Simplify and evaluate. Common expansions: sin x = x - x^3/6 + ..., e^x = 1 + x + x^2/2 + ..., log(1+x) = x - x^2/2 + ...

Check Continuity at a Point

Pattern

Determine whether a piecewise or composite function is continuous at a given point

How to recognize

Piecewise-defined function, function with different definitions on different intervals, or asking to find value of constant for continuity

→ Compute left-hand limit, right-hand limit, and f(a). If all three are equal, the function is continuous. For finding constants, set the limits equal and solve.

Check Differentiability at a Point

Pattern

Determine if a function is differentiable at a point, especially for |x|, piecewise, or max/min functions

How to recognize

Function involves absolute value, floor/ceiling, max/min, or is piecewise-defined. Question asks about differentiability at a boundary point.

→ Compute left-hand derivative (LHD) and right-hand derivative (RHD) at the point. If LHD = RHD and the function is continuous there, it is differentiable. Remember: differentiability implies continuity, but not vice versa.

Find Derivative from First Principles

Pattern

Compute f'(x) using the limit definition f'(x) = lim(h->0) [f(x+h) - f(x)]/h

How to recognize

Question explicitly asks for derivative using first principles, limit definition, or ab initio method

→ Substitute into the definition. Expand f(x+h) algebraically. Simplify the difference quotient. Take the limit as h approaches 0. Use standard limits if trig or exponential functions appear.

Apply Rolle's or LMVT

Pattern

Verify conditions and apply Rolle's theorem or Lagrange's Mean Value Theorem

How to recognize

Question gives a function on [a,b] and asks to find c, verify theorem conditions, or prove existence of a root of f'(x) = k

→ First check all conditions: continuity on [a,b], differentiability on (a,b), and f(a)=f(b) for Rolle's. For LMVT, set f'(c) = [f(b)-f(a)]/(b-a) and solve for c. Ensure c lies in (a,b).

Limit Involving Sequences and Series

Pattern

Evaluate limits of sequences or summation expressions as n approaches infinity

How to recognize

Expression involves n approaching infinity, summation with n terms, or recursive sequences

→ Convert sum to integral using lim (1/n) sum f(r/n) = integral from 0 to 1 of f(x) dx. For recursive sequences, assume limit L exists and solve the resulting equation. Use Squeeze theorem when bounds are available.

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Trigonometric Functions

Application of Derivatives

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