Calculus (Limits, Integration, Multi-variable)

Syllabus Unit: Engineering Mathematics

Why it matters

Chapter overview

Calculus in GATE's Engineering Mathematics section spans single-variable limits and derivatives, multivariable partial derivatives and gradients, double/triple integrals for area, volume and mass, and Taylor-series-based optimisation — together these tools let engineers model rates of change and locate optimal operating points across every branch's core courses.

General Aptitude & Mathematics Engineering Mathematics 15% weightage

Live Animation1 / 4

1Approach x→aCheck LHL and RHL

2Compare to f(a)Continuity test

3Apply diff rulesProduct/quotient/chain

4Check derivative existenceCorners fail

1Limits, Continuity & Differentiation

A limit asks what a function approaches, not what it equals — the gap between the two is exactly what makes sin(x)/x at x=0 solvable despite being 0/0.

Learning Objectives

Evaluate limits using standard forms (sin x/x, (1+x)^(1/x), L'Hopital's rule for 0/0 and ∞/∞).
State the three conditions for continuity of f at x=a.
Differentiate using the product, quotient and chain rules, and identify points of non-differentiability.
Distinguish continuity from differentiability using the standard counterexample |x| at x=0.

Key Terms

LimitThe value a function approaches as its input approaches a point, whether or not the function is actually defined there.

L'Hopital's RuleFor a genuine 0/0 or ∞/∞ indeterminate form, lim f/g = lim f'/g', applicable repeatedly.

Continuityf is continuous at x=a if f(a) is defined, the limit at a exists, and the limit equals f(a).

Differentiabilityf is differentiable at x=a if its derivative (instantaneous rate of change) exists there; it always implies continuity, never the reverse.

A limit describes the trend of a function near a point, which is why lim(x→0) sin(x)/x = 1 is solvable even though direct substitution gives the meaningless 0/0.

Standard limits GATE expects instantly

lim(x→0) sin(x)/x = 1
lim(x→0) (1 − cos x)/x² = 1/2
lim(n→∞) (1 + 1/n)ⁿ = e
lim(x→0) (aˣ − 1)/x = ln a

L'Hopital's rule

For genuine 0/0 or ∞/∞ forms, lim f(x)/g(x) = lim f'(x)/g'(x), provided the latter limit exists; it can be applied repeatedly if the new ratio is still indeterminate.

Common MistakeL'Hopital only applies to true 0/0 or ∞/∞ forms — applying it directly to forms like ∞−∞ or 0×∞ without first algebraically converting them is a classic GATE trap.

Continuity at x=a requires all three: f(a) is defined, lim(x→a) f(x) exists (left-hand limit = right-hand limit), and that limit equals f(a). |x| is continuous everywhere but NOT differentiable at x=0, since its left-hand derivative (−1) and right-hand derivative (+1) disagree — proof that continuity does not imply differentiability.