Limits, Continuity and Differentiability

Limits, Continuity and Differentiability is one of the most heavily tested calculus topics in NDA Mathematics. From 2010 onwards, every paper has included questions on evaluating standard limits, identifying points of discontinuity, and testing whether a piecewise or modulus function is differentiable at a given point. Unlike algebra topics that reward memorised formulas alone, this chapter requires you to reason carefully about left-hand and right-hand behaviour — a habit that also pays dividends in Derivatives and their Applications and Indefinite and Definite Integration.

What This Topic Covers

The NDA syllabus groups three closely related ideas under this heading:

• Limits — the behaviour of \(f(x)\) as \(x\) approaches a value. Standard limits (\(\sin x / x\), exponential, logarithmic), indeterminate forms, and limits at infinity.
• Continuity — whether a function can be drawn without lifting the pen. Three conditions must hold simultaneously: the function is defined at the point, the limit exists at that point, and the limit equals the function value.
• Differentiability — whether the derivative exists at a point. This requires the left-hand and right-hand derivatives to be equal. Differentiability is a stricter condition than continuity.

PYQs also test the greatest integer function \([x]\), the modulus function \(|x|\), and piecewise-defined functions for all three properties. These function types produce the most exam questions.

Exam Pattern & Weightage

The table below summarises the types of limits, continuity, and differentiability questions seen in NDA papers from 2010 to 2018 based on the PYQ file for this topic.

Year	Paper	Sub-type tested
2010	I	Continuity of piecewise function at a point; greatest integer discontinuity
2010	II	Standard limit \(\lim (\cos ax - \cos bx)/x^2\); continuity value finding
2011	I	Continuity and differentiability of \(|x|+2^x\); \(\lim x \sin(1/x)\)
2012	I	Continuity of \(|x-3|\); limits of rational functions; differentiable for all real \(x\)
2012	II	Existence of \(\lim (\sin x)/x\); \(\lim (1-\cos x)/x^2\)
2013	I	Continuity conditions (statement-type); \(\lim (\sin x - \tan x)/x^3\); differentiability at a point
2013	II	Limit of rational expression; continuity of piecewise cubic
2014	I	Continuity value finding; three-part piecewise differentiability; \(\lim \sqrt{f(x)}/\sqrt{x}\)
2014	II	Continuous-at-0 problems; \(\lim \log_5(1+x)/x\); \(\lim (1^2+2^2+\dots+n^2)/n^3\)
2015	I	Continuity with constants \(a\) and \(b\); three-piece function continuity; \(\lim\) of \(G(x)\)
2015	II	Continuity of sin/cos piecewise; derivability at \(x=0\); \(f(\pi)\) for continuity
2016	I	Differentiability of \(|x-1|+x^2\); greatest integer and modulus function limits
2016	II	Continuity of three-part function; differentiability of \(g(x)\)
2017	I	Standard limit \(\lim (1+x)^{1/x}\); differentiability for all \(x\); left-hand derivative
2017	II	Differentiability of \(x(\sqrt{x}-\sqrt{x+1})\); set of points of differentiability; continuity of \(\ln\), \(\cos\) pieces
2018	I	\(f(0)\) for continuity; \(\lim (\sin x)/(\sin 2x)\); \(\lim (\sqrt{2x+3h}-\sqrt{2x})/(2h)\)
2018	II	Continuity of \((x^2-9)/(x^2-2x-3)\) at \(x=3\); modulus function differentiability

Core Concepts

Standard Limits

These four results appear repeatedly across NDA papers. Commit them to memory — every limit problem either uses one directly or reduces to one after algebraic manipulation.

A 2010-II PYQ asks $$\lim_{x\to 0}\dfrac{\cos ax - \cos bx}{x^2}.$$ Use the identity $$\cos P - \cos Q = -2\sin\!\left(\tfrac{P+Q}{2}\right)\sin\!\left(\tfrac{P-Q}{2}\right)$$ and the standard limit to get \(\dfrac{b^2-a^2}{2}\).

A 2014-II PYQ asks: \(\lim_{x\to 0}\dfrac{\log_5(1+x)}{x}\). Apply the logarithmic standard limit directly: the answer is \(\log_5 e\).

Continuity Conditions

A function \(f(x)\) is continuous at \(x=a\) if and only if three conditions all hold:

If LHL \(\ne\) RHL the function has a jump discontinuity. If LHL = RHL but the common value \(\ne f(a)\), it is a removable discontinuity — the "hole" type NDA loves to test by asking you to redefine \(f(a)\).

If any condition fails, the function is discontinuous at \(x=a\). A 2013-I PYQ tests this as a statement-match: "\(f(x)\) is continuous at \(x=a\) if \(\lim f(x)\) exists" — this is false because the limit must also equal \(f(a)\).

For piecewise functions, continuity at a boundary point means equating the left-hand and right-hand limits and setting them equal to the function value. A 2011-I PYQ establishes the value of \(k\) that makes a piecewise function continuous at \(x=1\).

Differentiability

The derivative of \(f\) at \(x=a\) exists when the left-hand derivative (LHD) equals the right-hand derivative (RHD):

This implication is tested directly in 2013-I and 2016-I as a statement-type question. \(f(x)=|x|\) is the canonical counterexample: continuous at \(x=0\) but not differentiable there because \(\text{LHD}=-1\) and \(\text{RHD}=+1\).

Differentiability of Common Functions

PYQ data reveals these patterns:

• \(f(x)=|x-c|\) is continuous everywhere, differentiable everywhere except at \(x=c\).
• \(f(x)=[x]\) (greatest integer) is discontinuous and not differentiable at every integer.
• \(f(x)=e^x\) is differentiable for all \(x\in\mathbb{R}\) (confirmed in 2014-I PYQ: \(e^x\) is differentiable at \(x=0\)).
• \(f(x)=x^n\) is differentiable for all \(x\) when \(n\in(1,\infty)\) — a 2017-I PYQ tests this directly.
• \(f(x)=\sin|x|\) is continuous but not differentiable at \(x=0\).

Chain Rule and Implicit Differentiation

Though the heaviest NDA questions on this topic focus on limits and continuity, differentiability questions often require computing the derivative to check LHD = RHD.

A 2018-I PYQ asks for \(\lim_{h\to 0}\dfrac{\sqrt{2x+3h}-\sqrt{2x}}{2h}\). Recognise this as the derivative of \(\sqrt{2x}\) using the definition. Differentiate: \(\dfrac{d}{dx}\sqrt{2x}=\dfrac{1}{\sqrt{2x}}\). The limit equals \(\dfrac{1}{2\sqrt{2x}}\), which matches option (a) \(\dfrac{1}{\sqrt{2x}}\).

Worked Examples

Each example below is drawn from the PYQ file. Work through the steps before checking the solution.

Example 1 — Continuity value for a piecewise function (2010-I)

Q: If \(f(x)=\dfrac{x(x-2)}{x^2-4}\) for \(x\ne 2\), and \(f\) is continuous at \(x=2\), find \(f(2)\).

• Factor: $$x^2-4=(x-2)(x+2)$$ So $$f(x)=\frac{x(x-2)}{(x-2)(x+2)}=\frac{x}{x+2} \quad \text{for } x\ne 2.$$
• Take the limit: $$\lim_{x\to 2}\frac{x}{x+2}=\frac{2}{2+2}=\frac{2}{4}=\frac{1}{2}.$$
• For continuity, \(f(2)\) must equal this limit: \(f(2)=\tfrac{1}{2}\).

Example 2 — Continuity/differentiability of \(|x-3|\) (2012-I)

Q: Statement 1: \(f(x)=|x-3|\) is continuous at \(x=3\). Statement 2: \(f(x)=|x-3|\) is differentiable at \(x=0\). Which is correct?

• At \(x=3\): $$\text{LHL}=\lim_{x\to 3^{-}}(3-x)=0;\quad \text{RHL}=\lim_{x\to 3^{+}}(x-3)=0;\quad f(3)=0.$$ All three equal — continuous at \(x=3\). Statement 1 is true.
• At \(x=0\): \(f(x)=|x-3|=3-x\) for \(x\) near \(0\). This is a linear function near 0, so the derivative is \(-1\) on both sides. Differentiable at \(x=0\). Statement 2 is also true.
• Answer: Both statements 1 and 2 are correct.

Example 3 — Continuity of a three-part function (2015-I)

Q: \(f(x)=ax-2\) for \(-2 < x < -1\); \(f(x)=-1\) for \(-1\le x\le 1\); \(f(x)=a+2(x-1)\) for \(1 < x < 2\). Find \(a\) for \(f\) to be continuous at \(x=-1\) and \(x=1\).

• At \(x=-1\): $$\text{LHL}=a(-1)-2=-a-2,\quad \text{Value}=-1.$$ Equate: \(-a-2=-1 \implies a=-1\).
• Check at \(x=1\) with \(a=-1\): \(\text{RHL}=-1+2(1-1)=-1\). Value \(=-1\). Matches — consistent.
• Answer: \(a=-1\).

Example 4 — Differentiability of \(|x-1|+x^2\) (2016-I)

Q: For \(f(x)=|x-1|+x^2\), which statement is correct?

• At \(x=0\): \(f(x)=(1-x)+x^2\) near \(x=0\). Derivative \(=-1+2x\), which is \(-1\) at \(x=0\) from both sides. Differentiable at \(x=0\).
• At \(x=1\): LHD — use \(f(x)=(1-x)+x^2\) for \(x < 1\). \(f'(x)=-1+2x \to\) at \(x=1\): \(-1+2=1\). RHD — use \(f(x)=(x-1)+x^2\) for \(x > 1\). \(f'(x)=1+2x \to\) at \(x=1\): \(1+2=3\). \(\text{LHD}\ne\text{RHD}\).
• Answer: \(f(x)\) is continuous but not differentiable at \(x=1\).

Example 5 — 0/0 form by factoring

Q: Evaluate $$\lim_{x\to 2}\frac{x^2-4}{x-2}$$.

• Direct substitution: $$\frac{4-4}{2-2}=\frac{0}{0}$$ — indeterminate, so factorise.
• $$x^2-4=(x-2)(x+2)$$. Cancel the $$(x-2)$$ factor: the function reduces to $$x+2$$ for $$x\ne 2$$.
• Now substitute: $$\lim_{x\to 2}(x+2)=4$$.
• Answer: 4. (Same trick applies to any $$\frac{x^n-a^n}{x-a}$$ form, which equals $$n\,a^{n-1}$$.)

Example 6 — sin(x)/x application via L'Hôpital

Q: Evaluate $$\lim_{x\to 0}\frac{1-\cos 2x}{x^2}$$.

• Direct substitution: $$\frac{1-1}{0}=\frac{0}{0}$$. Apply L'Hôpital's rule.
• Differentiate numerator: $$\frac{d}{dx}(1-\cos 2x)=2\sin 2x$$. Differentiate denominator: $$\frac{d}{dx}(x^2)=2x$$.
• New limit: $$\lim_{x\to 0}\frac{2\sin 2x}{2x}=\lim_{x\to 0}\frac{\sin 2x}{x}=2\cdot\lim_{x\to 0}\frac{\sin 2x}{2x}=2\cdot 1=2$$.
• Answer: 2.

Example 7 — Continuity at a junction point

Q: Find $$k$$ so that $$f(x)=\frac{x^2-4}{x-2}$$ for $$x\ne 2$$ and $$f(2)=k$$ is continuous at $$x=2$$.

• For continuity at $$x=2$$: $$\lim_{x\to 2}f(x)=f(2)=k$$.
• From Example 5, the limit equals $$4$$.
• Therefore $$k=4$$. (Removable-discontinuity type — the most frequent NDA continuity pattern.)

Example 8 — Limit of \(G(x)\) using derivative definition (2015-I)

Q: If \(G(x)=\sqrt{25-x^2}\), find \(\lim_{x\to 1}\dfrac{G(x)-G(1)}{x-1}\).

• Recognise this as \(G'(1)\) — the derivative of \(G\) at \(x=1\).
• $$G'(x)=\frac{d}{dx}(25-x^2)^{1/2}=\tfrac{1}{2}(25-x^2)^{-1/2}\cdot(-2x)=\frac{-x}{\sqrt{25-x^2}}.$$
• At \(x=1\): $$G'(1)=\frac{-1}{\sqrt{25-1}}=\frac{-1}{\sqrt{24}}=\frac{-1}{2\sqrt{6}}.$$
• Answer: \(-\dfrac{1}{2\sqrt{6}}\).

Exam Shortcuts (Pro-Tips)

Limits and continuity questions reward speed. The five shortcuts below collapse multi-step calculations into one-line answers — each has direct payoff in NDA PYQs.

Shortcut 1 — L'Hôpital's Rule for 0/0 and ∞/∞

If direct substitution gives $$\frac{0}{0}$$ or $$\frac{\infty}{\infty}$$, differentiate numerator and denominator separately, then re-substitute. Reapply until the indeterminate form clears.

Use case: $$\lim_{x\to 0}\frac{\sin x - x}{x^3}$$ — three L'Hôpital passes give $$-\tfrac{1}{6}$$ in under 30 seconds.

Shortcut 2 — The 1^∞ Form Trick

If $$\lim_{x\to a}[f(x)]^{g(x)}$$ has the form $$1^\infty$$, skip logarithms and use the one-shot formula:

Example: $$\lim_{x\to 0}(1+3x)^{1/x}=e^{\lim_{x\to 0}(3x)\cdot\tfrac{1}{x}}=e^{3}$$. The NDA 2017-I PYQ on $$\lim_{x\to 0}(1+x)^{1/x}=e$$ is the simplest instance.

Shortcut 3 — Highest-Power Rule for x → ∞

For a rational function $$\dfrac{ax^m+\dots}{bx^n+\dots}$$ as $$x\to\infty$$:

Shortcut 4 — Standard-Limit Recognition Table

If the limit is at $$x\to 0$$ and the expression matches one of the six standard forms, write the answer directly. No need to derive — these are memorisation gifts.

Six Must-Memorise Standard Limits $$\frac{\sin x}{x},\;\frac{\tan x}{x},\;\frac{e^x-1}{x},\;\frac{\log(1+x)}{x}\;\to\;1$$ $$\frac{a^x-1}{x}\to\log_e a \qquad \left(1+\tfrac{1}{x}\right)^{x}\to e \;\text{ as }\; x\to\infty$$

Shortcut 5 — Continuity Check in One Glance

For piecewise functions, the only test is LHL = f(a) = RHL. Set up these three quantities side-by-side; if any two differ, the function fails continuity. For "find the constant" type, equate the two expressions evaluated at the junction.

⚡ NDA Alert

Polynomials, $$\sin x$$, $$\cos x$$, and $$e^x$$ are continuous AND differentiable everywhere — no junction checks needed. The exam-trap functions are $$|x-a|$$ (continuous but not differentiable at $$x=a$$) and $$[x]$$ (neither continuous nor differentiable at any integer). Indeterminate forms $$0\times\infty$$ and $$\infty-\infty$$ must be transformed (multiply/divide, take common denominator) into $$\tfrac{0}{0}$$ or $$\tfrac{\infty}{\infty}$$ before L'Hôpital applies.

Common Question Patterns

After reviewing PYQs from 2010 to 2018, five recurring patterns dominate this chapter:

The Five Patterns NDA Repeats

• Find f(a) for continuity: A piecewise or rational function with a removable discontinuity — factorise, cancel, and take the limit. Appears in 2010-I, 2011-I, 2014-I, 2018-I.
• Find constant for continuity: A function with unknown constants a, b — equate limits at boundary points. Appears in 2011-I, 2014-II, 2015-I, 2016-II.
• Statement-type: continuous vs. differentiable: Two or more statements about a function — identify which are true. Dominant question type from 2012 to 2017. Key rule: differentiable ⟹ continuous, not vice versa.
• Standard limit evaluation: Direct application of \(\lim (\sin x)/x=1\), \(\lim (1-\cos x)/x^2=\tfrac{1}{2}\), or exponential/log forms. Appears in almost every paper.
• Limit at infinity: Divide numerator and denominator by highest power of \(x\), then apply standard results. A 2011-I question asks $$\lim_{x\to\infty}\!\left(\sqrt{x^2+ax+1}-\sqrt{a^2x^2+1}\right);$$ the answer is \(\tfrac{a}{2}\) after dividing by \(x\).

How NDA Tests This Topic

Statement-match questions (Which of the following statements is/are correct?) dominate. You rarely need to compute a full derivative from scratch — instead, NDA tests whether you can classify a function's behaviour at a specific point using the definitions of LHD and RHD. Practise writing "\(\text{LHD}=\ldots,\;\text{RHD}=\ldots,\;\text{LHD}\ne\text{RHD}\implies\) not differentiable" in a structured way to answer statement questions reliably.

Preparation Strategy

Follow this four-stage approach over 10–14 days:

Stage 1 (Days 1–3): Lock in the formulas. Write the four standard limits on a flash card and derive them from scratch once. Then practise 10 direct-substitution or standard-limit MCQs until you solve them in under 30 seconds each. Cross-reference with Sets, Relations and Functions — many limits questions involve functional composition.

Stage 2 (Days 4–6): Continuity of piecewise functions. For every piecewise function, write down the three conditions explicitly. Work through at least 6 examples where you must find a constant for continuity. Common boundary types: \(x=0\), \(x=1\), \(x=\pi/2\). Practise equating one-sided limits systematically.

Stage 3 (Days 7–9): Differentiability at special points. For modulus functions \(|x-c|\), practice computing LHD and RHD by substituting \(x=c\pm h\) and taking \(h\to 0\). Do the same for functions defined differently on two sides of a point. Remember the key cases: \(|x|\), \([x]\), \(x\sin(1/x)\), and \(x^n\) for non-integer \(n\).

Stage 4 (Days 10–14): PYQ mock practice. Attempt full sets of PYQs from this topic under timed conditions (90 seconds per question). For any question you get wrong, identify which of the five patterns it belongs to and re-drill that pattern. This topic directly feeds into Derivatives and their Applications and Differential Equations — a strong foundation here multiplies your marks.

⚡ NDA Alert

NDA papers from 2016 onward frequently use statement-type questions with three sub-statements (1, 2, 3) about continuity or differentiability of the same function. Eliminate wrong statements one by one rather than trying to verify all three simultaneously — it is faster and reduces errors under exam pressure.

Frequently Asked Questions

Is a function differentiable if it is continuous?

No. Continuity is necessary but not sufficient for differentiability. The classic example from NDA PYQs is \(f(x)=|x|\), which is continuous at \(x=0\) but not differentiable there because its left-hand derivative is \(-1\) and right-hand derivative is \(+1\). Differentiability requires both one-sided derivatives to exist and be equal.

What is the value of \(\lim (\sin x)/x\) as \(x\to 0\)?

The limit equals \(1\), provided \(x\) is in radians. This is the most frequently used standard limit in NDA papers. A direct consequence: \(\lim (\sin kx)/(kx)=1\) for any constant \(k\ne 0\), and \(\lim (\sin ax)/(bx)=a/b\).

How do you find the value of a constant that makes a function continuous?

Equate the left-hand limit and right-hand limit at the boundary point, and set them equal to the function's value at that point. For example, if \(f(x)=kx^3\) for \(x\le 1\) and \(f(x)=2\) for \(x > 1\), continuity at \(x=1\) requires \(k\cdot 1^3=2\), so \(k=2\). NDA tests this type every sitting.

Where is the greatest integer function \([x]\) discontinuous?

At every integer. At any integer \(n\), the left-hand limit is \(n-1\) and the right-hand limit is \(n\), so the two-sided limit does not exist. This is confirmed in multiple NDA PYQs (2010-I, 2014-II, 2015-I, 2016-I). It is also not differentiable at any integer.

Does \(\lim (1-\cos x)/x^2\) equal \(1/2\)?

Yes. This standard result appears in a 2012-II NDA PYQ. The derivation uses the identity \(1-\cos x=2\sin^2(x/2)\), giving $$\lim_{x\to 0}\frac{2\sin^2(x/2)}{x^2}=2\cdot\frac{1}{4}\cdot\lim_{x\to 0}\!\left[\frac{\sin(x/2)}{x/2}\right]^2=\frac{1}{2}.$$

Is \(\sin|x|\) differentiable at \(x=0\)?

No. \(\sin|x|\) is continuous everywhere, but at \(x=0\): \(\text{LHD}=\lim_{h\to 0^{-}}\sin|h|/h=\lim \sin(-h)/h=-1\), and \(\text{RHD}=\lim_{h\to 0^{+}}\sin(h)/h=+1\). Since \(\text{LHD}\ne\text{RHD}\), it is not differentiable at \(x=0\). The 2011-I PYQ on \(f(x)=|x|+2^x\) tests a similar idea.

How many questions from this topic appear in a typical NDA paper?

Based on PYQ data from 2010 to 2018, this topic (combined with functions) contributes roughly 5–8 questions per NDA paper. Of these, 2–3 are typically limits problems, 2–3 are continuity problems, and 1–2 are differentiability problems. The topic is one of the highest-yielding in the calculus section.

Related Topics

Next Step Derivatives and their Applications Calculus Indefinite and Definite Integration Calculus Differential Equations Foundation Sets, Relations and Functions Overview NDA Maths — All Topics Practice NDA Mock Tests