Calculus: Limits, Continuity, Derivatives & Maxima-Minima

Calculus is the backbone of GATE ME Engineering Mathematics — expect 2–3 marks directly, plus it underlies fluid, heat and vibration questions. Maxima–minima and limit evaluation are the perennial favourites.

Core toolkit

Limits: For 0/0 or ∞/∞ forms use L'Hôpital: lim f/g = lim f'/g'. Standard limits: (sin x)/x → 1, (1 − cos x)/x² → 1/2, (eˣ − 1)/x → 1, (1 + 1/n)ⁿ → e.

Continuity: f is continuous at a if lim_{x→a} f(x) = f(a) (left limit = right limit = value). Differentiable ⇒ continuous, but not the reverse (e.g. |x| at 0).

Maxima–minima: set f'(x) = 0 for stationary points; f''(x) > 0 → minimum, f''(x) < 0 → maximum, f''(x) = 0 → inconclusive (check sign change or higher derivatives). Always test domain endpoints too.

Key theorems: Rolle's (f(a)=f(b) ⇒ some c with f'(c)=0); Mean Value Theorem f'(c) = [f(b) − f(a)]/(b − a); Taylor series f(x) = f(a) + f'(a)(x−a) + f''(a)(x−a)²/2! + …

Exam Tricks & Tips

• 🎯 Try direct substitution first — only reach for L'Hôpital if you actually get 0/0 or ∞/∞.
• 🎯 Memorise the standard limits above; most 1-mark limit NATs collapse to one of them.
• 🎯 For closed-interval optimisation, always compare stationary points AND endpoints — the max often sits at an endpoint.
• 🎯 Second-derivative test failing (f''=0)? Fall back to the first-derivative sign-change test rather than guessing.
• 🎯 1^∞, 0·∞, ∞−∞ forms: convert by taking log or rewriting as a fraction before applying L'Hôpital.
• ❌ Common mistake: applying L'Hôpital to a limit that is NOT indeterminate (e.g. 2/0 or 5/3) — it gives a wrong answer; the rule only applies to 0/0 and ∞/∞.

Expected exam pattern

A 1-mark limit or continuity/differentiability check, and a 2-mark applied optimisation ("minimise the cost/area/energy" giving a real design context). Taylor/Maclaurin expansion appears indirectly in numerical-methods questions.

Quick recap

Substitute first; L'Hôpital only for genuine 0/0 or ∞/∞. Continuity = matching left/right limits and value; differentiable ⇒ continuous. Optimise via f'=0 with the f'' sign test, and never forget endpoints. Know MVT, Rolle, and the Taylor form.

Cover the answer, recall, then check. 11 cards on single-variable calculus for GATE ME.

Q1. When can L'Hôpital's rule be applied?
A1. Only for indeterminate forms 0/0 or ∞/∞: then lim f/g = lim f'/g'. Convert other forms (1^∞, 0·∞) first.

Q2. State three must-know standard limits.
A2. (sin x)/x → 1, (1 − cos x)/x² → 1/2, and (eˣ − 1)/x → 1 as x → 0.

Q3. Define continuity of f at x = a.
A3. Left limit = right limit = f(a). All three must exist and be equal.

Q4. Is a continuous function always differentiable? Give an example.
A4. No. |x| is continuous at x = 0 but not differentiable there (corner). Differentiability implies continuity, not the reverse.

Q5. How do you classify a stationary point where f'(x₀) = 0?
A5. f''(x₀) > 0 → local minimum; f''(x₀) < 0 → local maximum; f''(x₀) = 0 → inconclusive, use sign-change test.

Q6. In closed-interval optimisation, where can the global extremum occur?
A6. At interior stationary points OR at the interval endpoints — always check both.

Q7. State the Mean Value Theorem.
A7. If f is continuous on [a,b] and differentiable on (a,b), there exists c with f'(c) = [f(b) − f(a)]/(b − a).

Q8. State Rolle's theorem.
A8. If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then f'(c) = 0 for some c in (a,b).

Q9. Write the Taylor series of f about x = a.
A9. f(x) = f(a) + f'(a)(x−a) + f''(a)(x−a)²/2! + f'''(a)(x−a)³/3! + …

Q10. Evaluate lim_{x→0} (tan x − sin x)/x³.
A10. = 1/2. (Expand: tan x − sin x ≈ x³/2 for small x.)

Q11. What does an inflection point require?
A11. f''(x) = 0 with a change of concavity (sign change of f''). Curvature switches from concave-up to concave-down or vice versa.

Calculus is the language the rest of the ME paper is written in — bending-moment maxima, optimum cutting speed, minimum-weight design and boundary-layer growth all reduce to "set the derivative to zero." Master the mechanics here and you speed up everywhere.

Core concept: the derivative is the local rate of change (slope), the integral accumulates it back, and continuity is the guarantee that a limit equals the function value so these tools even apply.

Deep explanation

Beginner — limits and continuity

A limit asks what f(x) approaches, not what it equals. For 0/0 or ∞/∞ forms use L'Hôpital's rule: lim f/g = lim f'/g'. A function is continuous at a if lim(x→a) f(x) = f(a) (limit exists and matches the value). Differentiability ⇒ continuity, but not the reverse (|x| is continuous but not differentiable at 0).

Intermediate — differentiation toolkit

Reflex these:

• Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v − uv')/v².
• Chain: d/dx f(g(x)) = f'(g)·g'.
• Standard: d/dx(sin x)=cos x, (eˣ)'=eˣ, (ln x)'=1/x, (xⁿ)'=n xⁿ⁻¹.

Taylor series near a: f(a) + f'(a)(x−a) + f''(a)(x−a)²/2! + … underlies numerical methods and error estimates.

Advanced — maxima, minima and inflection

At a stationary point f'(x) = 0. Classify with the second-derivative test:

• f''(x) > 0 → local minimum (curve holds water),
• f''(x) < 0 → local maximum,
• f''(x) = 0 → inconclusive; check sign change of f' or higher derivatives (possible inflection).
  Always compare stationary values with domain endpoints for a global optimum — GATE optimisation problems frequently hide the true extremum at a boundary.

Worked example

A wire of length 40 cm is bent into a rectangle. Find the dimensions of maximum area.
Perimeter: 2(x + y) = 40 → y = 20 − x. Area A = x(20 − x) = 20x − x².
A' = 20 − 2x = 0 → x = 10. A'' = −2 < 0 → maximum. So x = 10, y = 10: a 10×10 square, area 100 cm². (The optimum of a fixed-perimeter rectangle is always a square.)

GATE relevance

Directly tested in the Maths block (limits/L'Hôpital, extrema) and reused constantly: maximum bending moment, optimum tool life/cutting speed (Taylor's tool-life optimisation), minimum potential energy, and rate problems throughout thermo and fluids.

Exam tricks & shortcuts

• Indeterminate 0/0? L'Hôpital first, don't factor laboriously.
• For "maximum area/volume for fixed perimeter/surface," expect a symmetric answer (square, cube, sphere).
• Mnemonic for concavity: "Min holds water (f''>0), Max sheds it (f''<0)."

Key formulas

• Standard limits: lim(x→0) sinx/x = 1; (eˣ−1)/x = 1; L'Hôpital for 0/0.
• Maxima/minima: f′(x) = 0; f″ < 0 max, f″ > 0 min.
• Two variables: f_x = f_y = 0; D = f_xx f_yy − f_xy² (D>0,f_xx<0 → max; D>0,f_xx>0 → min; D<0 → saddle).
• Taylor: f(x) = Σ fⁿ(a)(x−a)ⁿ/n!.
• Mean value theorem: f′(c) = [f(b)−f(a)]/(b−a).

Use the Hessian discriminant D for multivariable optimisation; L'Hôpital resolves indeterminate limits.

Worked Example

Problem: (a) Evaluate the limit lim_(x→0) (1 − cos x)/x². (b) Find the minimum value of f(x) = x + 1/x for x > 0.

Solution:
(a) This is a 0/0 indeterminate form. Use the small-angle (Taylor) expansion cos x ≈ 1 − x²/2 + …, so 1 − cos x ≈ x²/2:
lim_(x→0) (1 − cos x)/x² = lim_(x→0) (x²/2)/x² = 1/2.
(Alternatively, apply L'Hôpital's rule twice: (sin x)/(2x) → cos x / 2 → 1/2.)

(b) Minimise f(x) = x + 1/x for x > 0.
f'(x) = 1 − 1/x². Set f'(x) = 0:
1 − 1/x² = 0 ⇒ x² = 1 ⇒ x = 1 (taking the positive root since x > 0).
Second derivative test: f''(x) = 2/x³, and f''(1) = 2 > 0 ⇒ local minimum.
Minimum value: f(1) = 1 + 1 = 2.

Answer: (a) 1/2; (b) minimum value 2 at x = 1.