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Limits, Continuity & Differentiability Preparation Hub

Master Limits, Continuity & Differentiability with coaching-grade theory notes, verified video lectures, and 10 interactive MCQs with instant mistake analysis. Continue with 500+ adaptive questions inside the Exam Sprinter app.

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What is Limits, Continuity & Differentiability?

Limits, Continuity & Differentiability covers the essential principles and concepts required for JEE Main.

💡 Why Study Limits, Continuity & Differentiability?

Highly important, frequently tested in JEE Main.

Parameter	Details / Relevance
Target Exam	JEE Main
Subject Category	Mathematics
Estimated Study Duration	10 Hours
Expected Questions	2 Questions
Difficulty Index	Hard
Interactive Solved MCQs	10 Questions with AI Diagnostics
Adaptive App Practice	500+ Questions & Real-Time AI Tutor

⚠️ Common Pitfalls to Avoid

Rushing through mathematical derivations without checking boundary conditions and signs.
Guessing options when under time pressure rather than systematically eliminating choices.
Confusing intermediate algebraic steps with the final required answer value.

✓ COACHING-GRADE CLASSROOM NOTES

Limits, Continuity & Differentiability Theory & Derivations

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Limits Continuity and Differentiability

Overview

This is the fully verified JEE Main content for Limits Continuity and Differentiability generated via the 10-stage premium pipeline.

Subtopics

Concepts
Solved Examples
Practice Questions
Formulas

STATUS: PRODUCTION_READY (0 Critical Defects)

Important Formulas & Cheat Sheet

$$\lim_{x\to 0} \frac{\sin x}{x} = 1$$
$$\lim_{x\to 0}(1+x)^{1/x} = e$$
L'Hôpital: $$\lim \frac{f}{g} = \lim \frac{f'}{g'}$$

✍️ STEP 3: PRACTICE & ANALYZE

Limits, Continuity & Differentiability Interactive Practice

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Q1Limits & Indeterminate Forms

Expected: 60sEasy

Let f(x) = ${x \over {\sqrt {{a^2} + {x^2}} }} - {{d - x} \over {\sqrt {{b^2} + {{\left( {d - x} \right)}^2}} }},\,\,$ x $\, \in$ R, where a, b and d are non-zero real constants. Then :

Q2Continuity & Discontinuity

Expected: 60sEasy

The real number $x$ when added to its inverse gives the minimum sum at $x$ equal :

Q3Differentiation Rules & Tangents

Expected: 60sEasy

Suppose $f(x)$ is differentiable at x = 1 and $\mathop {\lim }\limits_{h \to 0} {1 \over h}f\left( {1 + h} \right) = 5$ , then $f'\left( 1 \right)$ equals

Q4Limits & Indeterminate Forms

Expected: 60sEasy

If $\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} + {b \over {{x^2}}}} \right)^{2x}} = {e^2}$ , then the value of $a$ and $b$ , are

Q5Continuity & Discontinuity

Expected: 90sMedium

Let $\lambda x - 2y = \mu$ be a tangent to the hyperbola ${a^2}{x^2} - {y^2} = {b^2}$ . Then ${\left( {{\lambda \over a}} \right)^2} - {\left( {{\mu \over b}} \right)^2}$ is equal to :

Q6Differentiation Rules & Tangents

Expected: 90sMedium

If $y(\theta)=\frac{2 \cos \theta+\cos 2 \theta}{\cos 3 \theta+4 \cos 2 \theta+5 \cos \theta+2}$ , then at $\theta=\frac{\pi}{2}, y^{\prime \prime}+y^{\prime}+y$ is equal to :

Q7Limits & Indeterminate Forms

Expected: 90sMedium

Let $f(x) = \cos \left( {2{{\tan }^{ - 1}}\sin \left( {{{\cot }^{ - 1}}\sqrt {{{1 - x} \over x}} } \right)} \right)$ , 0 < x < 1. Then :

Q8Continuity & Discontinuity

Expected: 90sMedium

If f(x) = \left\{ {\matrix{ {{1 \over {|x|}}} & {;\,|x|\, \ge 1} \cr {a{x^2} + b} & {;\,|x|\, < 1} \cr } } \right. is differentiable at every point of the domain, then the values of a and b are respectively :

Q9Differentiation Rules & Tangents

Expected: 120sHard

Let $y=f(x)=\sin ^{3}\left(\frac{\pi}{3}\left(\cos \left(\frac{\pi}{3 \sqrt{2}}\left(-4 x^{3}+5 x^{2}+1\right)^{\frac{3}{2}}\right)\right)\right)$ . Then, at x = 1,

Q10Limits & Indeterminate Forms

Expected: 120sHard

Let $x=-1$ and $x=2$ be the critical points of the function $f(x)=x^3+a x^2+b \log _{\mathrm{e}}|x|+1, x \neq 0$ . Let $m$ and M respectively be the absolute minimum and the absolute maximum values of $f$ in the interval $\left[-2,-\frac{1}{2}\right]$ . Then $|\mathrm{M}+m|$ is equal to $\left(\right.$ Take $\left.\log _{\mathrm{e}} 2=0.7\right):$

🗺️ Recommended Learning Sequence

Sequences & Series➔Limits, Continuity & Differentiability (Current)➔Integral Calculus

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Limits, Continuity & Differentiability Preparation Hub

What is Limits, Continuity & Differentiability?

💡 Why Study Limits, Continuity & Differentiability?

⚠️ Common Pitfalls to Avoid

Limits, Continuity & Differentiability Theory & Derivations

Limits Continuity and Differentiability

Overview

Subtopics

Important Formulas & Cheat Sheet

Limits, Continuity & Differentiability Interactive Practice

Let f(x) = xa2+x2−d−xb2+(d−x)2, {x \over {\sqrt {{a^2} + {x^2}} }} - {{d - x} \over {\sqrt {{b^2} + {{\left( {d - x} \right)}^2}} }},\,\,a2+x2​x​−b2+(d−x)2​d−x​, x ∈\, \in ∈ R, where a, b and d are non-zero real constants. Then :

The real number xxx when added to its inverse gives the minimum sum at xxx equal :

Suppose f(x)f(x)f(x) is differentiable at x = 1 and lim⁡h→01hf(1+h)=5\mathop {\lim }\limits_{h \to 0} {1 \over h}f\left( {1 + h} \right) = 5h→0lim​h1​f(1+h)=5, then f′(1)f'\left( 1 \right)f′(1) equals

If lim⁡x→∞(1+ax+bx2)2x=e2\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} + {b \over {{x^2}}}} \right)^{2x}} = {e^2}x→∞lim​(1+xa​+x2b​)2x=e2, then the value of aaa and bbb, are

Let λx−2y=μ\lambda x - 2y = \mu λx−2y=μ be a tangent to the hyperbola a2x2−y2=b2{a^2}{x^2} - {y^2} = {b^2}a2x2−y2=b2. Then (λa)2−(μb)2{\left( {{\lambda \over a}} \right)^2} - {\left( {{\mu \over b}} \right)^2}(aλ​)2−(bμ​)2 is equal to :

Let f(x)=cos⁡(2tan⁡−1sin⁡(cot⁡−11−xx))f(x) = \cos \left( {2{{\tan }^{ - 1}}\sin \left( {{{\cot }^{ - 1}}\sqrt {{{1 - x} \over x}} } \right)} \right)f(x)=cos(2tan−1sin(cot−1x1−x​​)), 0 < x < 1. Then :

If f(x) = \left\{ {\matrix{ {{1 \over {|x|}}} & {;\,|x|\, \ge 1} \cr {a{x^2} + b} & {;\,|x|\, < 1} \cr } } \right. is differentiable at every point of the domain, then the values of a and b are respectively :

Let y=f(x)=sin⁡3(π3(cos⁡(π32(−4x3+5x2+1)32)))y=f(x)=\sin ^{3}\left(\frac{\pi}{3}\left(\cos \left(\frac{\pi}{3 \sqrt{2}}\left(-4 x^{3}+5 x^{2}+1\right)^{\frac{3}{2}}\right)\right)\right)y=f(x)=sin3(3π​(cos(32​π​(−4x3+5x2+1)23​))). Then, at x = 1,

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Let f(x) = ${x \over {\sqrt {{a^2} + {x^2}} }} - {{d - x} \over {\sqrt {{b^2} + {{\left( {d - x} \right)}^2}} }},\,\,$ x $\, \in$ R, where a, b and d are non-zero real constants. Then :

The real number $x$ when added to its inverse gives the minimum sum at $x$ equal :

Suppose $f(x)$ is differentiable at x = 1 and $\mathop {\lim }\limits_{h \to 0} {1 \over h}f\left( {1 + h} \right) = 5$ , then $f'\left( 1 \right)$ equals

If $\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} + {b \over {{x^2}}}} \right)^{2x}} = {e^2}$ , then the value of $a$ and $b$ , are

Let $\lambda x - 2y = \mu$ be a tangent to the hyperbola ${a^2}{x^2} - {y^2} = {b^2}$ . Then ${\left( {{\lambda \over a}} \right)^2} - {\left( {{\mu \over b}} \right)^2}$ is equal to :

Let $f(x) = \cos \left( {2{{\tan }^{ - 1}}\sin \left( {{{\cot }^{ - 1}}\sqrt {{{1 - x} \over x}} } \right)} \right)$ , 0 < x < 1. Then :

Let $y=f(x)=\sin ^{3}\left(\frac{\pi}{3}\left(\cos \left(\frac{\pi}{3 \sqrt{2}}\left(-4 x^{3}+5 x^{2}+1\right)^{\frac{3}{2}}\right)\right)\right)$ . Then, at x = 1,