JEE Main Maths Limits, Continuity & Differentiability 2027: Standard Limits & Maxima-Minima

This chapter is the engine room of JEE Main Calculus. Limits set the foundation, continuity and differentiability decide whether a function behaves nicely, and derivatives turn that behaviour into concrete answers: where a function is rising, where it peaks, and how fast a quantity changes.

A function can be continuous without being differentiable. Think of a sharp corner: no break, but no single well-defined slope either.

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Chapter at a Glance

Priority reflects past-paper trends, not an official NTA weightage.

What the NTA Syllabus Covers

• Real-valued functions, algebra of functions, polynomial, rational, trigonometric, logarithmic and exponential functions, inverse functions, graphs of simple functions
• Limits, continuity and differentiability
• Differentiation of sum, difference, product and quotient of functions, and of trigonometric, inverse trigonometric, logarithmic, exponential, composite and implicit functions, derivatives up to order two
• Applications of derivatives: rate of change of quantities, monotonic increasing and decreasing functions, maxima and minima of functions of one variable

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Master These Topics

1. Standard Limits Worth Memorising

$$\lim_{x \to 0} \frac{\sin x}{x} = 1 \qquad \lim_{x \to 0} \frac{e^x - 1}{x} = 1 \qquad \lim_{x \to 0} \frac{\ln(1+x)}{x} = 1 \qquad \lim_{x \to 0} (1+x)^{1/x} = e$$

Worked example: $\displaystyle\lim_{x \to 0} \frac{\sin 3x}{x} = \lim_{x \to 0} 3 \cdot \frac{\sin 3x}{3x} = 3$.

Worked example: $\displaystyle\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^{x} = e^{3}$, using the standard exponential limit form.

Trap: Both standard limit and L'Hopital-style tricks apply only to indeterminate forms like $0/0$ or $\infty/\infty$. Check the form before applying any shortcut.

2. Continuity and Differentiability

A function is continuous at $x = a$ when the left-hand limit, right-hand limit and the function value all agree: $\displaystyle\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$.

Differentiability at a point requires the left-hand derivative to equal the right-hand derivative there. Every differentiable function is continuous, but the reverse is false.

Worked example: $f(x) = \lvert x \rvert$ is continuous at $x = 0$, but its left-hand derivative is $-1$ and its right-hand derivative is $+1$, so it is not differentiable there. The graph has a sharp corner instead of a smooth turn.

3. Differentiation Rules

• Product rule: $(uv)' = u'v + uv'$
• Quotient rule: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$
• Chain rule: $\dfrac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

Worked example: For $y = \sin(x^2)$, the chain rule gives $\dfrac{dy}{dx} = \cos(x^2) \cdot 2x$.

4. Applications: Increasing, Decreasing, Maxima and Minima

A function is increasing where $f'(x) \gt 0$ and decreasing where $f'(x) \lt 0$. At a local extremum, $f'(x) = 0$; the second derivative test then tells you which kind: $f''(x) \lt 0$ gives a local maximum, and $f''(x) \gt 0$ gives a local minimum.

Worked example: For $f(x) = x^3 - 3x$, $f'(x) = 3x^2 - 3 = 0$ gives $x = \pm 1$. Since $f''(x) = 6x$, we get $f''(-1) = -6 \lt 0$ (local maximum, value $2$) and $f''(1) = 6 \gt 0$ (local minimum, value $-2$). The function increases on $(-\infty, -1)$ and $(1, \infty)$, and decreases on $(-1, 1)$.

Worked example (rate of change): The radius of a circular ripple grows at $2\text{ cm/s}$. When $r = 5\text{ cm}$, the area grows at

$$\frac{dA}{dt} = 2\pi r \frac{dr}{dt} = 2\pi (5)(2) = 20\pi \text{ cm}^2/\text{s}$$

Trap: $f'(x) = 0$ only locates candidates for extrema. Always confirm with the second derivative test or a sign change of $f'$, since it could also be a point of inflection.

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Common Traps to Avoid

• Applying a standard limit to a form that is not $0/0$.
• Assuming continuity guarantees differentiability.
• Forgetting the chain rule factor when differentiating a composite function.
• Treating every point with $f'(x) = 0$ as a maximum or minimum without checking further.

60-Second Revision Sheet

• $\lim_{x\to0}\dfrac{\sin x}{x} = 1$, $\lim_{x\to0}\dfrac{e^x-1}{x} = 1$, $\lim_{x\to0}(1+x)^{1/x} = e$
• Continuous at $a$: LHL $=$ RHL $= f(a)$; differentiable needs LHD $=$ RHD
• Product $u'v+uv'$; quotient $\tfrac{u'v-uv'}{v^2}$; chain $f'(g(x))g'(x)$
• Increasing where $f' \gt 0$; local max needs $f'=0, f''\lt0$; local min needs $f'=0, f''\gt0$

Your Study Plan

1. Day 1: standard limits and indeterminate forms.
2. Day 2: continuity and differentiability, including piecewise functions.
3. Day 3: differentiation rules for all standard function types.
4. Day 4: increasing/decreasing functions, maxima, minima and rate of change, timed set.

Practice Limit, Continuity & Differentiability Questions Free → (opens in a new tab)

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Continue Your Maths Journey

• Previous chapter: Sequence & Series
• Next chapter: Integral Calculus
• All 14 JEE Main Maths chapters
• Complete JEE Main Syllabus 2027 guide

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