Important Limit and Continuity & Differentiability of Function Formulas for JEE and NEET

Things To Remember :

Limit - Definitions and Fundamental Results

1. Existence of a limit. The limit of a function f(x) as x → a is said to exist and be finite when the left-hand and right-hand limits are equal and finite. In symbolic form:

2. Fundamental theorems on limits. Let g(x) → m and f(x) → l as x → a. If l and m exist (finite), then the following hold:

• 
• 
• Provided m ≠ 0:
  
• 
  and for any constant k, k·f(x) = k
  
  f(x).
• If f is continuous at g(x) = m then
  
  = f(m).

For example:

Standard limits and common forms

3. Standard limits.

• 
  where x is measured in radians.
• 
  (note also the related result)
  
  (1 - h)ⁿ → 0 as h → 0 for fixed n.
• If f(x) → 1 and Ф(x) → ∞, then
  
• If f(x) → A > 0 and Ф(x) → B (finite) then
  
  = e^z where z =
  
  , hence ln[f(x)] → e^{B ln A} = A^B.
• 
  = ln a (a > 0). In particular
  
• 

Squeeze (Sandwich) Theorem

4. Squeeze Theorem. If f(x) ≤ g(x) ≤ h(x) for all x near a (except possibly at a) and

with

and hence lim_x→a g(x) = l.

Indeterminate forms and useful notes

5. Indeterminate forms.

Note.

• Infinity (∞) is a symbol, not a number; it cannot be plotted and does not obey elementary algebra rules.
• ∞ + ∞ = ∞
• ∞ × ∞ = ∞
• If a is finite then a/∞ = 0.
• a/0 is not defined if a ≠ 0.
• For finite a and b, ab = 0 if and only if a = 0 or b = 0.

Strategies for evaluating limits

6. Useful methods to evaluate limits.

• Factorisation.
• Rationalisation or double rationalisation.
• Use of trigonometric identities, appropriate substitutions and standard limits.
• Series expansions: binomial, exponential, logarithmic, and series expansions of sin x, cos x, tan x. These expansions are important and should be memorised where appropriate.

The common expansions (as referenced) are:

Continuity

Things To Remember :

1. Definition of continuity at a point. A function f(x) is said to be continuous at x = c if

that is, the left-hand limit and the right-hand limit at x = c exist, are equal and equal to f(c):

In words: LHL at x = c = RHL at x = c = f(c).

Continuity at x = a is meaningful only if the function is defined in an immediate neighbourhood of a (it need not be defined at a for the limit to exist).

2. Reasons for discontinuity.

• 
  The limit does not exist at the point.
• The function is not defined at x = c.
• 
  f(x) exists but f(c) is different from the common limit.

Geometrically, such points appear as breaks or jumps in the graph. For example, a typical graph may be discontinuous at x = 1, 2 and 3.

Types of discontinuities

3. Type-1: Removable discontinuity. If lim_x→c f(x) exists but is not equal to f(c) (or f(c) is not defined), the discontinuity is removable. We can redefine f(c) suitably to make f continuous at c.

Removable discontinuities include:

• Missing point discontinuity: lim exists finitely but f(a) is not defined.
  
  Example: f(x) =
  
  has a missing point at x = 1.
• Isolated point discontinuity: lim exists and f(a) exists but lim ≠ f(a).
  
  Example: f(x) =
  
  for x ≠ 4 and f(4) = 9 has an isolated discontinuity at x = 4. The function [x] + [-x] has isolated discontinuities at all integer x.

Type-2: Non-removable discontinuity. If lim_x→c f(x) does not exist then the discontinuity cannot be removed by redefining f(c). These include:

• Finite jump discontinuity: the left and right limits exist and are finite but unequal. Example: f(x) = x - [x] at integral x.
• Infinite discontinuity: one or both one-sided limits are infinite. Examples involve functions like 1/(x - a) at x = a.
• Oscillatory discontinuity: the function oscillates without approaching a limit (e.g., sin(1/x) as x → 0).

In all cases of non-removable discontinuity, lim_x→a f(x) does not exist.

Note. In the adjacent example graph:

• f is continuous at x = -1.
• f has an isolated discontinuity at x = 1.
• f has a missing point discontinuity at x = 2.
• f has a non-removable (finite type) discontinuity at the origin.

4. Jump of discontinuity and piecewise continuity. The absolute finite difference between the RHL and LHL at x = c is called the jump of discontinuity. A function with finitely many jumps in an interval I is called piecewise continuous or sectionally continuous on that interval.

5. Continuity of common functions. All polynomials, trigonometric, exponential and logarithmic functions are continuous on their domains.

6. Arithmetic of continuous functions. If f and g are continuous at x = c, then:

• f ± g and k·f (for any constant k) are continuous at x = c.
• f·g is continuous at x = c.
• If g(c) ≠ 0 then f/g is continuous at x = c; equivalently:
  

7. Intermediate value theorem. If f is continuous on a closed interval I and a, b ∈ I, then for any y₀ between f(a) and f(b) there exists c ∈ (a, b) such that f(c) = y₀. Consequently:

• If f(a) and f(b) have opposite signs, there exists at least one root of f(x) = 0 in (a, b).
• If K is any real number between f(a) and f(b), there exists at least one solution to f(x) = K in (a, b).

Important remarks about products and continuity.

• If f is continuous and g is discontinuous at x = a, the product φ(x) = f(x)·g(x) may or may not be discontinuous at a (see example
  
  ).
• If both f and g are discontinuous at x = a, their product may still be continuous (see example
  
  ).
• Point (indicator) functions are treated as discontinuous. Example: f(x) =
  
  is not continuous at x = 1.
• A continuous function on a closed domain has a range that is also an interval (closed if the domain is closed and function is continuous).
• If f is continuous at x = c and g is continuous at f(c), then the composite g∘f is continuous at x = c. Examples of such continuous functions at x = 0 are given by
  
  and their composite:
  

7. Continuity in an interval.

• (a) f is continuous in (a, b) if it is continuous at every point of (a, b).
• (b) f is continuous in the closed interval [a, b] if:
  1. f is continuous in (a, b),
  2. f is right-continuous at a, i.e.
     
     = f(a) (finite),
  3. f is left-continuous at b, i.e.
     
     = f(b) (finite).

Functions continuous on [a, b] satisfy the properties given above under the Intermediate Value Theorem.

8. Single point continuity. Functions that are continuous at only one point are said to have single point continuity. Example: f(x) =

Differentiability

Things To Remember :

1. Right-hand and left-hand derivatives. By definition, the derivative at x = a (if it exists) is

(i) The right-hand derivative f′(a⁺) is defined by:

provided the limit exists and is finite.

(ii) The left-hand derivative f′(a^-) is defined by:

provided the limit exists and is finite.

We also write f′(a⁺) = f′₊(a) and f′(a^-) = f′_-(a).

Geometrically, the existence of a finite derivative at x = a means a unique tangent of finite slope can be drawn at that point.

Derivability and continuity

(iii) Relation between differentiability and continuity.

If f′(a) exists then f is differentiable at x = a ⇒ f is continuous at x = a.

Sketch of reasoning (informal):

For f′(x) =

Note: If f is differentiable at every point of its domain, then it is continuous on that domain.

The converse is not true: continuity does not imply differentiability. For example, functions like f(x) =

Important observations:

• Let f′₊(a) = p and f′_-(a) = q, where p and q are finite. If p = q, f is differentiable at a and hence continuous at a.
• If p ≠ q then f is not differentiable at a, although it may still be continuous there.
• In short:
  • Differentiability ⇒ Continuity
  • Continuity ⇏ Differentiability
  • Non-differentiability ⇏ Discontinuity
  • But: Discontinuity ⇒ Non-differentiability
• If f is continuous but not differentiable at a, the graph typically has a sharp corner at a.

Derivability over an interval

3. Derivability on an interval. A function f is said to be differentiable over an interval if it is differentiable at each point of the interval. For the closed interval [a, b], f is differentiable on [a, b] if:

• f′(a⁺) and f′(b^-) exist, and
• for every c with a < c < b, f′₊(c) and f′_-(c) exist and are equal.

Notes and algebraic properties of derivatives.

• If f and g are differentiable at x = a, then f ± g and f·g are differentiable at x = a.
• If g(a) ≠ 0, then f/g is differentiable at x = a.
• If f is differentiable at x = a and g is not, the product f·g can still be differentiable at a (example: f(x) = x and g(x) = |x|).
• If both f and g are not differentiable at a, their difference or sum may still be differentiable (examples: f(x) = |x| and g(x) = -|x|).
• Derivability of f at a does not necessarily imply f′ is continuous at a.

6. A useful result (product with zero value). Suppose f and g are defined on an interval containing x₀. If f is differentiable at x₀ with f(x₀) = 0 and g is continuous at x₀, then F(x) = f(x)·g(x) is differentiable at x₀. Example: F(x) = sin x · x^2/3 is differentiable at x = 0.

Worked Examples (short, step-wise style)

Example 1. Evaluate lim_x→0 (sin x)/x.

Sol.

Using the standard trigonometric limit:

Therefore lim_x→0 (sin x)/x = 1.

Example 2. If f(x) = x·sin(1/x) for x ≠ 0 and f(0) = 0, show f is continuous at x = 0 but not differentiable there.

Sol.

For continuity:

For x ≠ 0, |x·sin(1/x)| ≤ |x|.

As x → 0, |x| → 0, so x·sin(1/x) → 0 = f(0). Hence f is continuous at 0.

For differentiability:

Consider the difference quotient:

[f(h) - f(0)]/h = [h·sin(1/h) - 0]/h = sin(1/h).

As h → 0 this oscillates between -1 and 1 and does not have a limit. Hence f′(0) does not exist.

Example 3. Show that if f′(a) exists then f is continuous at a.

Sol.

By definition f′(a) = lim_h→0 [f(a + h) - f(a)]/h exists (finite).

Then lim_h→0 [f(a + h) - f(a)] = lim_h→0 h·[f(a + h) - f(a)]/h = 0.

Therefore lim_x→a f(x) = f(a). Hence f is continuous at a.