Limits Continuity Differentiation Questions for AGNIVEER VAYU

Limits and Limit Laws

A limit describes the value a function approaches as the input approaches some value. Think of it like aiming at a target without necessarily hitting it. Limit laws let you break complex limits into simpler pieces. Key insight: the limit exists and equals L if the left-hand limit (approaching from negative side) and right-hand limit (approaching from positive side) both equal L. In Agniveer papers, you'll often see limits that look undefined at first (0/0 form) but can be simplified by factoring, rationalizing, or using L'Hopital's rule.

• lim(x→a) [f(x) + g(x)] = lim f(x) + lim g(x) (sum rule)
• lim(x→a) [f(x) × g(x)] = lim f(x) × lim g(x) (product rule)
• lim(x→a) [f(x) / g(x)] = lim f(x) / lim g(x), provided lim g(x) ≠ 0
• If lim(x→a) f(x) = L, then lim(x→a) [f(x)]^n = L^n (power rule)
• Sandwich theorem: if g(x) ≤ f(x) ≤ h(x) and lim g(x) = lim h(x) = L, then lim f(x) = L

Continuity at a Point

A function is continuous at a point if three conditions hold: (1) f(a) is defined, (2) lim(x→a) f(x) exists, and (3) lim(x→a) f(x) = f(a). Imagine drawing the graph without lifting your pen—that's continuity. Agniveer exams often ask: 'Find the value of constant k such that f(x) is continuous at x = a.' The trick is to set left-hand limit equal to right-hand limit equal to f(a), then solve for k.

• Continuous at a point means no jump, hole, or vertical asymptote at that point
• Check left-hand limit: lim(x→a-) f(x)
• Check right-hand limit: lim(x→a+) f(x)
• If left-hand limit ≠ right-hand limit, function is discontinuous (jump discontinuity)
• Polynomial, exponential, sine, cosine functions are continuous everywhere on their domain
• Rational functions are discontinuous where denominator = 0

Differentiation: Definition and Rules

The derivative measures the rate of change of a function—the slope of the tangent line at a point. The formal definition is f'(a) = lim(h→0) [f(a+h) - f(a)]/h. In practice, you'll use derivative rules to avoid this tedious limit calculation. Agniveer questions test your ability to apply product rule, quotient rule, and chain rule quickly. A common trap: students forget the chain rule when differentiating composite functions like sin(x^2) or e^(3x).

• Power rule: d/dx [x^n] = n × x^(n-1)
• Sum rule: d/dx [f(x) + g(x)] = f'(x) + g'(x)
• Product rule: d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x) (remember: first times derivative of second, plus second times derivative of first)
• Quotient rule: d/dx [f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2
• Chain rule: d/dx [f(g(x))] = f'(g(x)) × g'(x) (derivative of outer times derivative of inner)
• d/dx [e^x] = e^x, d/dx [ln(x)] = 1/x, d/dx [sin(x)] = cos(x), d/dx [cos(x)] = -sin(x)

Differentiability and Relationship to Continuity

If a function is differentiable at a point, it must be continuous at that point. However, the reverse is not always true: a function can be continuous but not differentiable (e.g., at a sharp corner or cusp). In Agniveer exams, you may see a piecewise function and be asked to check both continuity and differentiability. The key: check left and right derivatives separately. If left derivative ≠ right derivative, the function is not differentiable at that point.

• Differentiability implies continuity, but continuity does not imply differentiability
• Left derivative: lim(h→0-) [f(a+h) - f(a)]/h
• Right derivative: lim(h→0+) [f(a+h) - f(a)]/h
• Function is differentiable at a point if left and right derivatives are equal
• Absolute value function |x| is continuous everywhere but not differentiable at x=0 (sharp corner)

• Forgetting to check both left-hand and right-hand limits when testing continuity; many students check only one side
• Misapplying L'Hopital's rule to forms other than 0/0 or ∞/∞; it does NOT work for 0×∞ or ∞-∞ without rewriting first
• Dropping the chain rule: students differentiate e^(3x) as e^(3x) instead of 3e^(3x), or sin(x^2) as cos(x^2) instead of 2x×cos(x^2)
• In quotient rule, writing (u'v - uv')/v instead of (u'v - uv')/v^2; forgetting to square the denominator
• Assuming a function is differentiable everywhere just because it is continuous; forgetting to check for corners, cusps, or vertical tangents

• LIATE for choosing u in integration by parts (also helps remember which function to differentiate first in product rule): Logarithm, Inverse trig, Algebraic, Trigonometric, Exponential
• For quotient rule: 'Low d-high minus high d-low, square the bottom and away we go'
• Chain rule: 'Outside in, inside out'—differentiate the outside function, then multiply by the derivative of the inside
• Continuity checklist: DEF (Defined, Exists limit, Function value equals limit)

• Agniveer Vayu papers typically include 1-2 limit problems (often involving algebraic simplification or L'Hopital's rule) worth 3-4 marks each
• Continuity questions almost always ask you to find a constant k; set up the equation lim(x→a-) = lim(x→a+) = f(a) and solve—this is a formula-based approach
• Differentiation questions test product rule, chain rule, and quotient rule in combination; a single problem may require all three
• Watch for piecewise functions: examiners love asking 'Is f(x) continuous and differentiable at x=a?' Answer both parts separately
• Time management: limit and continuity problems are usually quicker (4-5 mins); differentiation can take 6-8 mins if it involves multiple rules—prioritize accordingly