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Coal India Management Trainee (CIL MT)

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6.3 Limits, Continuity, Differentiability & Mean Value Theorems

Q: The function f(x) = |x| at x = 0 is:

Continuous but not differentiable. The left and right limits both equal 0 and match f(0), so the function is continuous. However the left-hand derivative is -1 while the right-hand derivative is +1, so the derivative does not exist. This is the standard demonstration that continuity does not imply differentiability.

Q: For f(x) = x^2 - 4x + 3 on the interval [1, 3], the value of c guaranteed by Rolle's theorem is:

Both f(1) and f(3) equal zero, so Rolle's theorem applies. Setting the derivative 2c - 4 equal to zero gives c = 2, which lies inside the open interval (1, 3).

Q: Which condition is NOT required for Rolle's theorem to apply on [a, b]?

f is monotonically increasing on [a, b]. Rolle's theorem requires only continuity on the closed interval, differentiability on the open interval, and equal values at the endpoints. Monotonicity is never required — indeed a function satisfying f(a) = f(b) and strictly monotonic on the interval cannot exist.

Q: The limit of (sin x)/x as x approaches 0, with x measured in radians, equals:

One. This is a standard limit with value 1 when x is expressed in radians. The value pi/180 arises only if x is measured in degrees, which is a frequent trap in objective papers.

Key Takeaways

A function is continuous at a point when the left-hand limit, right-hand limit and functional value are all equal and finite; differentiability additionally requires equal left- and right-hand derivatives.
Differentiability implies continuity, but continuity does not imply differentiability — the modulus function at x = 0 is the standard counter-example.
Rolle's theorem needs continuity on the closed interval, differentiability on the open interval, and equal end values, and then guarantees a point where the derivative vanishes.
Lagrange's mean value theorem drops the equal-end-value condition and guarantees a point where the derivative equals the average slope (f(b) - f(a))/(b - a).

Last updated: August 2026

Limits: The Working Definition

The CIL paper never asks for an epsilon-delta proof. What it asks is whether a limit exists and what its value is. The operational test is simple: the limit exists at $x = a$ when the left-hand and right-hand limits are finite and equal.

\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L \quad \Longrightarrow \quad \lim_{x \to a} f(x) = L

The standard limits that must be memorised:

Limit	Value
$\lim_{x\to0} \dfrac{\sin x}{x}$	$1$
$\lim_{x\to0} \dfrac{\tan x}{x}$	$1$
$\lim_{x\to0} \dfrac{1 - \cos x}{x^2}$	$1/2$
$\lim_{x\to0} \dfrac{e^x - 1}{x}$	$1$
$\lim_{x\to0} \dfrac{\ln(1+x)}{x}$	$1$
$\lim_{x\to0} (1 + x)^{1/x}$	$e$
$\lim_{x\to\infty} \left(1 + \dfrac{1}{x}\right)^{x}$	$e$

Note that $\sin x / x \to 1$ only when $x$ is in radians. A question phrased in degrees is a deliberate trap; the answer becomes $\pi/180$ .

Continuity

A function $f$ is continuous at $x = a$ when three conditions hold simultaneously:

$f(a)$ is defined,
$\lim_{x\to a} f(x)$ exists,
$\lim_{x\to a} f(x) = f(a)$ .

Discontinuities are classified as removable (the limit exists but disagrees with $f(a)$ , so redefining one point repairs it), jump (left and right limits both exist but differ), and infinite (at least one one-sided limit is unbounded).

A practical example from mining instrumentation: a step change in a conveyor load cell signal is a jump discontinuity; a single corrupted sample in an otherwise smooth trace is removable.

Differentiability

The derivative at a point is the two-sided limit

f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}

and it exists only when the left-hand derivative equals the right-hand derivative. The critical implication chain, tested almost every year in PSU papers, runs one way only:

\text{differentiable} \Longrightarrow \text{continuous}, \qquad \text{continuous} \nRightarrow \text{differentiable}

The canonical counter-example is $f(x) = |x|$ at $x = 0$ . It is continuous there, but the left-hand derivative is $-1$ and the right-hand derivative is $+1$ , so no derivative exists. Geometrically, any sharp corner or cusp destroys differentiability while leaving continuity intact.

Rolle's Theorem

If $f$ is (i) continuous on the closed interval $[a, b]$ , (ii) differentiable on the open interval $(a, b)$ , and (iii) satisfies $f(a) = f(b)$ , then there exists at least one $c \in (a,b)$ with

f'(c) = 0

Geometrically: a smooth curve that starts and ends at the same height must be horizontal somewhere in between. All three hypotheses are needed — dropping any one produces a valid counter-example, and CIL-style items often ask precisely which hypothesis fails.

Example. For $f(x) = x^2 - 4x + 3$ on $[1, 3]$ : $f(1) = 0$ and $f(3) = 0$ , the function is a polynomial so continuity and differentiability are automatic. Then $f'(c) = 2c - 4 = 0$ gives $c = 2$ , which indeed lies in $(1,3)$ .

Lagrange's Mean Value Theorem

Drop the condition $f(a) = f(b)$ and the conclusion generalises: there exists $c \in (a,b)$ such that

f'(c) = \frac{f(b) - f(a)}{b - a}

The instantaneous rate of change somewhere in the interval equals the average rate of change across it. Rolle's theorem is simply the special case in which the right-hand side is zero.

A physical reading familiar from mine haulage: if a dumper covers 30 km in 30 minutes, its average speed is 60 km/h, and the mean value theorem guarantees that its speedometer read exactly 60 km/h at some instant.

Example. For $f(x) = x^2$ on $[1, 4]$ , the average slope is $(16 - 1)/(4 - 1) = 5$ . Setting $f'(c) = 2c = 5$ gives $c = 2.5 \in (1,4)$ .

Cauchy's Mean Value Theorem

The generalisation to two functions states that if $f$ and $g$ satisfy the same hypotheses and $g'(x) \neq 0$ on $(a,b)$ , then for some $c$ ,

\frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)}

Setting $g(x) = x$ recovers Lagrange's form. Cauchy's theorem is the formal justification for L'Hopital's rule, which is treated in the next section under indeterminate forms.

Quick Reference: Which Theorem Applies

Given	Use	Conclusion
$f(a) = f(b)$	Rolle	$f'(c) = 0$
$f(a) \neq f(b)$	Lagrange	$f'(c) = $ average slope
Ratio of two functions	Cauchy	Ratio of derivatives = ratio of increments

Test Your Knowledge

The function f(x) = |x| at x = 0 is:

Continuous but not differentiable

Both continuous and differentiable

Neither continuous nor differentiable

Differentiable but not continuous

Test Your Knowledge

For f(x) = x^2 - 4x + 3 on the interval [1, 3], the value of c guaranteed by Rolle's theorem is:

1.5

2.5

Test Your Knowledge

Which condition is NOT required for Rolle's theorem to apply on [a, b]?

f is differentiable on the open interval (a, b)

f(a) = f(b)

f is monotonically increasing on [a, b]

f is continuous on the closed interval [a, b]

Test Your Knowledge

The limit of (sin x)/x as x approaches 0, with x measured in radians, equals:

Infinity

Pi divided by 180

Zero

One

Sections you finish are checked off in the contents.

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6.4 Maxima-Minima, Indeterminate Forms & Taylor Series

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