Chapter 13 — Limits And Derivatives

Fig 1 — Educational Diagram: Intuitive idea of limit

What is a limit (intuitively)?
The limit of a function f(x) as x approaches a point a is the value L that the outputs f(x) get closer and closer to, when x gets arbitrarily close to a (from either side). The actual value f(a) may be equal to L, different from L, or not even defined — but the limit concerns only how f(x) behaves near a.

Informal picture: Imagine walking along the curve y = f(x) toward the vertical line x = a. If your height y approaches a fixed number L no matter how you approach x = a (from left or right), then lim_x→a f(x) = L.

Left and right limits: A left-hand limit lim_x→a− f(x) looks at values when x approaches a from values less than a; a right-hand limit lim_x→a+ f(x) looks from values greater than a. The (two-sided) limit exists iff both one-sided limits exist and are equal.

Other behaviours:

• Removable discontinuity: f(x) approaches L but f(a) is different or undefined (a "hole").
• Jump discontinuity: left and right limits exist but are different → overall limit does not exist.
• Infinite limit / vertical asymptote: f(x) grows without bound as x→a.
• Limit at infinity / horizontal asymptote: as x→∞ (or −∞), f(x) approaches a constant.

How to find limits (basic intuitive methods): try direct substitution first. If that gives a finite number, that is the limit. If you get an indeterminate form (0/0), try algebraic simplification — factorization, cancellation, rationalization, trig identities — or use known standard limits (e.g. sin x / x → 1 as x→0), the squeeze theorem, or compare growth rates for limits at infinity.

Connection to rigorous definition: Formally, lim_x→a f(x)=L means: for every small tolerance ε>0 there exists a δ>0 such that whenever 0<|x−a|<δ, then |f(x)−L|<ε. (For class 11 it is enough to understand this idea informally.)

Fig 2 — Educational Diagram: Limits of functions — basic definitions

Intuitive idea: The limit of a function f(x) as x approaches a number a is the value that f(x) gets closer and closer to when x gets closer and closer to a (but x need not equal a). We write this as lim_x→a f(x) = L.

One-sided limits: The left-hand limit (approach from values less than a) is written lim_x→a⁻ f(x). The right-hand limit (approach from values greater than a) is written lim_x→a⁺ f(x). The (two-sided) limit lim_x→a f(x) exists and equals L iff both one-sided limits exist and are equal: lim_x→a⁻ f(x) = lim_x→a⁺ f(x) = L.

Limits can be finite or infinite: If f(x) grows without bound as x→a, we say lim_x→a f(x) = ∞ (or −∞) — this indicates a vertical asymptote or blow-up. If x→∞ (or −∞) we study behaviour at infinity; if f(x) approaches a finite number M, we write lim_x→∞ f(x) = M (horizontal asymptote).

Formal (epsilon–delta) definition — compact form: lim_x→a f(x) = L means: for every ε>0 there exists δ>0 such that 0 < |x−a| < δ implies |f(x)−L| < ε. (This formalises the idea of f(x) getting arbitrarily close to L when x is sufficiently close to a.)

Relation to continuity: f is continuous at a if lim_x→a f(x) exists and equals f(a). If the limit exists but differs from f(a), there is a removable discontinuity (a hole).

Useful facts and techniques: Direct substitution works for polynomials and many rational functions (when denominator ≠ 0). When direct substitution gives 0/0, simplify algebraically (factor, cancel) or use standard limit rules. The squeeze (sandwich) theorem helps when a function is trapped between two others with the same limit.

Fig 3 — Educational Diagram: Algebra and properties of limits

What is a limit (intuitive): The limit of f(x) as x approaches a value a is the value that f(x) gets arbitrarily close to when x is taken arbitrarily close to a (from either side), regardless of the value f(a) actually takes. We write this as lim(x→a) f(x) = L.

Why algebra of limits matters: Limits allow us to handle expressions that become indeterminate (like 0/0) by manipulating algebraically and using known limit rules. They are the foundation for derivatives and continuity.

Basic (algebraic) properties / limit laws:

• If lim(x→a) f(x) = A and lim(x→a) g(x) = B (finite), then
  • Sum: lim(f+g) = A + B
  • Difference: lim(f−g) = A − B
  • Scalar multiple: lim(c f) = c A for constant c
  • Product: lim(f g) = A B
  • Quotient: lim(f/g) = A/B provided B ≠ 0
  • Power/root: lim([f]^n) = A^n (for integer n), and lim( (f)^(1/n) ) = A^(1/n) when A is in the domain of the root
• Polynomials and rational functions: For a polynomial p(x), lim(x→a) p(x) = p(a). For a rational function r(x) = p(x)/q(x), if q(a) ≠ 0, lim(x→a) r(x) = p(a)/q(a). If you get 0/0, try algebraic simplification (factorization, cancellation, conjugate) to remove the indeterminacy.
• Squeeze (Sandwich) theorem: If h(x) ≤ f(x) ≤ g(x) for all x near a (except possibly at a) and lim(x→a) h(x) = lim(x→a) g(x) = L, then lim(x→a) f(x) = L. This is useful for oscillatory functions multiplied by a factor tending to 0.
• Limit of a composition: If lim(x→a) g(x) = L and f is continuous at L, then lim(x→a) f(g(x)) = f(L).
• Order/inequality property: If f(x) ≤ g(x) near a and both limits exist, then lim f(x) ≤ lim g(x).

How to compute limits algebraically (common techniques):

• Direct substitution when expression is defined at a.
• Factorization and cancellation when you get 0/0.
• Multiply by conjugate when radicals produce 0/0.
• Use squeeze theorem for oscillatory terms multiplied by something tending to 0.
• Use known limits and limit laws to combine simpler limits.

Link to continuity and derivatives: A function f is continuous at a if lim(x→a) f(x) = f(a). Many limit algebra rules are simply consequences of continuity of basic functions (polynomials, rational functions where denominator ≠ 0, roots, trig functions at points where defined, exponential and logarithm where defined).

Fig 4 — Educational Diagram: Techniques and standard limits

What this topic covers
Techniques and standard limits explain how to evaluate limits of functions as the variable approaches a finite value (including removable/discontinuous points) or infinity. You learn algebraic/special techniques and memorize a few standard limits used as building blocks.

Basic limit laws (used throughout)
If lim_{x->a} f(x)=L and lim_{x->a} g(x)=M (finite), then

• lim (f+g) = L+M
• lim (cf) = cL for constant c
• lim (fg) = LM
• lim (f/g) = L/M provided M ≠ 0

Common algebraic techniques

• Direct substitution: If f is continuous at a, lim_{x->a} f(x) = f(a).
• Factorization / cancel common factors: Useful when expression gives 0/0. Factor numerator/denominator and cancel the (x−a) factor.
• Rationalization (multiply by conjugate): For roots, multiply numerator and denominator by conjugate to remove indeterminate form.
• Divide by highest power (limits at infinity): For rational functions, divide numerator and denominator by the highest power of x to find limit as x→±∞.
• Use standard limits / small-angle approximations: Replace expressions near 0 by their limits (eg. sin x ≈ x) to evaluate more complex forms.
• Squeeze (sandwich) theorem: If g(x) ≤ f(x) ≤ h(x) near a and lim g = lim h = L, then lim f = L. Useful for trig limits.

Important standard limits (frequently used)

• lim_{x->0} (sin x)/x = 1
• lim_{x->0} (1 - cos x)/x^2 = 1/2
• lim_{x->0} (e^x - 1)/x = 1
• lim_{x->0} ln(1+x)/x = 1
• lim_{x->0} (1 + x)^{1/x} = e
• More generally, lim_{x->0} (1 + ax)^{1/x} = e^a

Remarks

• Many limits give an indeterminate form like 0/0 or ∞/∞; algebraic manipulation or standard limits resolve them.
• L'Hôpital's rule is a powerful technique for 0/0 or ∞/∞ but is usually introduced later (Class 12). In Class 11 you should rely on algebraic methods and the standard limits above.

How to approach a limit problem — stepwise

1. Attempt direct substitution.
2. If indeterminate, try factorization or cancel common terms.
3. For expressions with roots, try rationalization.
4. For trig expressions near 0, use standard trig limits or squeeze theorem.
5. For infinity, compare degrees (highest power) or use division by highest power.
6. If still hard, try rewriting using exponentials/logarithms for forms like (1+… )^{…}.

Fig 5 — Educational Diagram: Continuity of a function

Definition: A function f is said to be continuous at a point a if the limit of f(x) as x approaches a exists and equals the value of the function at that point. In symbols: f is continuous at a iff lim_x→a f(x) = f(a). If this holds for every point of an interval, f is continuous on that interval.

Left and right continuity: Continuity at a requires both one-sided limits to exist and be equal to f(a): lim_x→a⁻ f(x) = lim_x→a⁺ f(x) = f(a). If only the left (or right) limit equals f(a), f is left-continuous (or right-continuous) at a.

Why the limit must equal f(a): For continuity we need three things: (1) f(a) is defined, (2) lim_x→a f(x) exists, and (3) the two are equal. If any fails, the function is discontinuous at a.

Types of discontinuities:

• Removable discontinuity: limit exists but is not equal to f(a) or f(a) is not defined (a hole in the graph). It can be "fixed" by redefining f(a).
• Jump (or step) discontinuity: left and right limits exist but are unequal (graph has a jump).
• Infinite (essential) discontinuity: the function grows without bound near a (vertical asymptote) or one-sided limit is infinite.

Continuity rules and common continuous functions: If f and g are continuous at a, then f+g, f−g, f·g are continuous at a; f/g is continuous at a provided g(a) ≠ 0. Polynomials are continuous everywhere; rational functions are continuous where denominator ≠ 0; trig, exponential and log functions are continuous on their natural domains. Composition: if g is continuous at a and f is continuous at g(a), then f∘g is continuous at a.

Intuitive (epsilon–delta) idea: Continuity means small changes in x near a produce small changes in f(x) near f(a). Formally: for every ε>0 there exists δ>0 such that |x−a|<δ ⇒ |f(x)−f(a)|<ε.

Use in calculus: Continuity is essential for evaluating limits, applying the Intermediate Value Theorem (a continuous function on [a,b] takes every value between f(a) and f(b)), and for differentiability (differentiability implies continuity).

Fig 6 — Educational Diagram: Derivative — definition and interpretation

Definition (limit form)
The derivative of a function f at a point a is the limit of the difference quotient if it exists:

f'(a) = lim_{h->0} (f(a+h) - f(a)) / h

Equivalently,

f'(a) = lim_{x->a} (f(x) - f(a)) / (x - a)

Geometric interpretation
The derivative f'(a) is the slope of the tangent line to the curve y = f(x) at the point (a, f(a)). Geometrically this is the limit of slopes of secant lines through (a, f(a)) and (a+h, f(a+h)) as h -> 0.

Physical/Rate interpretation
If y = f(t) is a physical quantity depending on t (for example displacement), then f'(t) is the instantaneous rate of change (for displacement it is instantaneous velocity). The derivative gives how fast y changes with respect to x at a point.

Differentiability and continuity
If f is differentiable at a then f is continuous at a. The converse need not hold (example: f(x)=|x| is continuous at 0 but not differentiable there). Differentiability at a means the left-hand and right-hand difference-quotient limits are equal:

lim_{h->0+} (f(a+h)-f(a))/h = lim_{h->0-} (f(a+h)-f(a))/h

Local linear approximation (tangent line)
When f is differentiable at a, near x = a the function is approximated by its tangent line:

f(x) ≈ f(a) + f'(a)(x - a)

This gives the best linear estimate of f near a.

Use for monotonicity
If f'(x) > 0 on an interval, f is increasing there; if f'(x) < 0, f is decreasing. Zeros of f' often indicate local maxima/minima (use further tests to classify).

Fig 7 — Educational Diagram: Basic differentiation rules

What is a derivative?
The derivative of a function f at x is the instantaneous rate of change of f with respect to x. It is defined by the limit:
f'(x) = lim_{h→0 (f(x+h) − f(x))/h. If this limit exists, f is differentiable at x. Differentiability implies continuity, but not vice versa.

Why rules?
The limit definition is fundamental but often hard to use directly. Basic differentiation rules let us find derivatives quickly for many functions by algebraic manipulation.

Basic rules (intuitive summary)

• Constant rule: derivative of a constant is 0 (a function that does not change has zero rate of change).
• Power rule: for x^n, f'(x)=n x^{n−1} (proved using binomial expansion for integer n; extends to rational/real n under domain conditions).
• Constant multiple rule: d/dx[c·f(x)] = c·f'(x).
• Sum/Difference rule: d/dx[f±g] = f' ± g'.
• Product rule: d/dx[f·g] = f'·g + f·g'. Use when two functions multiply.
• Quotient rule: d/dx[f/g] = (f'·g − f·g') / g^2 (g ≠ 0).
• Chain rule (for composite functions): d/dx[f(g(x))] = f'(g(x)) · g'(x).

Standard derivatives (useful to memorize): d/dx[ e^x ] = e^x, d/dx[a^x] = a^x ln a, d/dx[ln x] = 1/x (x>0), d/dx[sin x] = cos x, d/dx[cos x] = −sin x, d/dx[tan x] = sec^2 x.

Notes & cautions: The power rule x^n for non-integer n requires x in a domain where x^n is defined and differentiable (e.g., x>0 for real non-integer powers). Functions like |x| are continuous everywhere but not differentiable at x=0 (corner/cusp).

Fig 8 — Educational Diagram: Derivatives of standard functions

What is a derivative?
The derivative of a function f at x, denoted f'(x), is the instantaneous rate of change of f with respect to x. It is defined by the limit:

f'(x) = lim_{h->0} [f(x+h) - f(x)] / h

This limit, when it exists, gives the slope of the tangent to the curve y = f(x) at the point (x, f(x)).

Common (standard) functions and their derivatives
Below are derivatives you will use frequently in Class 11. Domain restrictions are given where relevant.

• Constant: if f(x) = c (constant), then f'(x) = 0.
• Power rule: if f(x) = x^n (n any real number where expression is defined), then f'(x) = n x^{n-1}.
• Exponential: d/dx[e^x] = e^x. For a^x (a>0, a≠1): d/dx[a^x] = a^x ln a.
• Logarithm: d/dx[ln x] = 1/x (x>0). More generally, d/dx[log_a x] = 1/(x ln a) (x>0).
• Trigonometric functions: d/dx[sin x] = cos x, d/dx[cos x] = -sin x, d/dx[tan x] = sec^2 x (where defined). Also d/dx[cot x] = -csc^2 x, d/dx[sec x] = sec x tan x, d/dx[csc x] = -csc x cot x.
• Inverse trigonometric (useful — check syllabus): d/dx[arcsin x] = 1 / sqrt(1-x^2) for |x|<1; d/dx[arctan x] = 1/(1+x^2) for all real x; d/dx[arccos x] = -1 / sqrt(1-x^2).

Useful derivative rules (short reminder): derivatives are linear: (af + bg)' = a f' + b g'. For products and quotients use product and quotient rules. For composite functions use the chain rule.

Interpretation & intuition
The derivative measures how fast the output changes when the input changes a little. Geometrically it is the slope of the tangent line. Physically it often represents rates: velocity is the derivative of position; marginal cost is the derivative of total cost with respect to output.

Notes on domains and continuity: A derivative exists only where the limit definition is satisfied. For example ln x and 1/x are defined only for x>0 or x≠0 respectively; tan x and sec x have vertical asymptotes where cosine is zero.

Fig 9 — Educational Diagram: Chain rule and composite functions

What is a composite function?
If u = g(x) and y = f(u), then y is a composite function of x via u; written y = f(g(x)) or y = (f \u25cf g)(x). The inner function is g and the outer function is f.

Chain rule (informal statement)
If g is differentiable at x and f is differentiable at u = g(x), then the composite f(g(x)) is differentiable at x and

(f \u25cf g)'(x) = f'(g(x)) \cdot g'(x).

Why it works (short justification)
Let u = g(x). The change in y when x changes by Δx is Δy ≈ f'(u)·Δu and Δu ≈ g'(x)·Δx. Multiplying gives Δy ≈ f'(g(x))·g'(x)·Δx, so dy/dx = f'(g(x))·g'(x). A rigorous proof uses limits: limit of [f(g(x+Δx))−f(g(x))]/Δx = f'(g(x))·g'(x).

Extensions

• For several nested functions y = f(g(h(x))) the derivative is y' = f'(g(h(x)))·g'(h(x))·h'(x).
• Chain rule combines with other rules: e.g., d/dx [F(x)^n] = n·F(x)^{n-1}·F'(x); d/dx [ln(g(x))] = g'(x)/g(x).

Conditions
Both inner and outer functions must be differentiable at the relevant points (outer differentiable at the value of the inner). If the outer derivative exists only as a one-sided derivative that must be checked accordingly.

Geometric view
If at x the inner function changes with slope g'(x) and the outer function changes with slope f'(u) where u=g(x), then slopes multiply to give the slope of the composition: slope_y = slope_f_at_u × slope_g_at_x.

Fig 10 — Educational Diagram: Differentiability — exceptions and examples

What differentiability means: A function f is differentiable at a point x=a if the limit

f'(a) = lim_{h->0} (f(a+h) - f(a))/h

exists as a finite number. If this limit exists, we call it the derivative at a and f is differentiable at a.

Relation with continuity: If f is differentiable at a, then f is continuous at a. The converse is not true: continuity does not guarantee differentiability.

One-sided derivatives and the check for differentiability: The derivative exists at a iff the left-hand derivative

f'_-(a) = lim_{h->0-} (f(a+h)-f(a))/h

and the right-hand derivative

f'_+(a) = lim_{h->0+} (f(a+h)-f(a))/h

both exist (finite) and f'_-(a) = f'_+(a). If they differ or one is infinite or does not exist, f is not differentiable at a.

Common types of exceptions (points where differentiability fails)

• Corner or corner point: left and right derivatives are finite but unequal. Example shape: absolute value at 0.
• Cusp: left and right derivatives are infinite with opposite signs (slope tends to +infinity on one side and -infinity on the other). Example: y = x^{2/3} at 0.
• Vertical tangent: left and right derivatives both tend to ±infinity with the same sign (slope vertical). Example: y = x^{1/3} at 0.
• Discontinuity: if f is not continuous at a, it cannot be differentiable there. Example: greatest-integer (floor) function at integers.
• Oscillation: the difference quotient fails to approach a limit because of rapid oscillation (e.g., functions like x sin(1/x) at 0 give nonexistence of derivative when appropriately defined).

Practical check: To test differentiability at a point, compute the left and right difference-quotient limits. If both finite and equal, f is differentiable; otherwise it is not.

Differentiability on intervals and at endpoints: For an open interval (a,b), use the standard two-sided derivative. At an endpoint of a closed interval, differentiability requires the appropriate one-sided derivative to exist and be finite.

Fig 11 — Educational Diagram: Problem-solving techniques and applications (basic)

Overview: This topic introduces practical techniques to evaluate limits and derivatives and shows basic applications (rate, tangent, approximation, simple optimization). You learn methods to simplify expressions, find derivatives using rules or first principles, and apply derivatives to interpret instantaneous rates and local behaviour of functions.

Core ideas:

• Limit techniques: Use algebraic simplification (factorization/cancellation), rationalization (multiply by conjugate), trigonometric identities, series/expansion for small terms, substitution when continuous, and the squeeze theorem for trapped expressions. Recognize and handle removable (holes), jump, and infinite discontinuities.
• Derivative concept: Derivative f'(a) is the instantaneous rate of change or slope of tangent at x = a. Definition (first principle): f'(a) = lim_{h->0} (f(a+h)-f(a))/h, when the limit exists.
• Derivative rules (problem solving): power rule, sum/difference, product, quotient, and chain rules let you compute derivatives quickly after simplification. Use derivative tables for common elementary functions (polynomial, trig, exponential, logarithm).
• Applications (basic): tangent line equation, linear approximation (local linearisation), instantaneous rates (velocity from position), identifying increasing/decreasing intervals and simple maxima/minima via f'(x)=0 or sign changes, and small-change approximations (Δy ≈ f'(a)Δx).

Step-by-step problem-solving approach:

1. For limits: (a) Try direct substitution. (b) If 0/0 or ∞/∞ form, simplify: factor and cancel common factors; rationalize square roots; use trig identities (e.g. sin2x = 2sinx cosx); use standard limits like sin x/x → 1 as x→0. (c) Apply squeeze theorem when expression is bounded between two simpler limits.
2. For derivatives: (a) If asked from first principle, set up the difference quotient and simplify algebraically before taking the limit. (b) For routine differentiation, simplify expression (expand if needed), then apply rules (product, quotient, chain). (c) Check dimension/units for applied problems (e.g., velocity = derivative of displacement w.r.t time).
3. For applications: (a) Compute f'(x) and find critical points where f' = 0 or undefined. (b) Use sign chart for f' or second derivative test (f''>0 => local min; f''<0 => local max). (c) Use tangent line for approximations: f(x) ≈ f(a) + f'(a)(x-a).

Tips: Always simplify algebraically before taking limits. Sketch or visualize the function when possible. Keep track of domain and points where function or derivative is undefined. For modelling problems, state what variable depends on what (e.g., y(t)).