Unit 2: Functions, limit and continuity - Subjective Questions

A function from a set to a set is a rule that assigns to each element exactly one element . It is denoted by .

• Domain: The set of all inputs is called the domain of .
• Co-domain: The set into which the function maps is called the co-domain.
• Range: The set of all actual output values is called the range. The range is always a subset of the co-domain.

Example: Let be defined by .

• Domain (all real numbers)
• Co-domain
• Range since squares of real numbers are non-negative.

1. Identity Function:

Defined as for all .

• Domain , Range .
• Its graph is a straight line passing through the origin at to the x-axis.

2. Constant Function:

Defined as , where is a fixed real number.

• Domain , Range .
• Its graph is a horizontal line parallel to the x-axis at height .

3. Modulus (Absolute Value) Function:

Defined as:

• Domain , Range .
• Its graph is V-shaped with vertex at the origin.

Signum Function:

Defined as:

• Domain , Range .
• It indicates the sign of the input value.

Greatest Integer Function (Floor Function):

Defined as = the greatest integer less than or equal to .

• Domain , Range (set of integers).
• Examples: , , .
• Its graph is a step function.

For two real functions and with the same domain , the algebra of functions is defined as:

• Sum:
• Difference:
• Product:
• Quotient: , provided
• Scalar multiple:

Given and :

Limit of a Function:

The limit of as approaches is , written as:

if can be made arbitrarily close to by taking sufficiently close to (but not equal to ).

Left-Hand Limit (LHL):

The value approached as tends to from values less than :

Right-Hand Limit (RHL):

The value approached as tends to from values greater than :

The limit exists only if .

Example: For at :

• RHL
• LHL

Since LHL RHL, the limit does not exist at .

Let and . Then the following rules hold:

• Sum Rule:
• Difference Rule:
• Product Rule:
• Quotient Rule: , provided
• Scalar Multiple Rule:
• Power Rule:

These theorems allow complex limits to be broken into simpler limits that can be evaluated individually.

Direct substitution gives , an indeterminate form. We factor the numerator.

Using with :

Therefore:

Cancel the common factor :

Substitute :

Answer:

Proof:

Consider the expression . Factoring the numerator:

There are terms in the second bracket. Dividing:

Taking the limit as , each of the terms tends to :

Application: For , here , :

Answer:

Method for Rational Functions:

A rational function is where and are polynomials.

• First, try direct substitution of .
• If , the limit is simply .
• If both and (form ), factor both numerator and denominator, cancel the common factor , then substitute.

Evaluation:

At : numerator , denominator . Form is .

Factor:

• Numerator:
• Denominator:

Answer:

Standard Trigonometric Limits:

Proof of (Sandwich Theorem):

For a small positive angle (in radians), by comparing areas in a unit circle:

Dividing throughout by (positive):

Taking reciprocals (reversing inequalities):

As , . By the Sandwich Theorem, since is squeezed between and :

(a)

Multiply and divide to use the standard limit :

(b)

Use the identity :

Answers: (a) , (b)

Continuity at a Point:

A function is said to be continuous at if:

This single equation requires three conditions to hold simultaneously:

• 1. is defined — the function has a value at (i.e., is in the domain).
• 2. exists — the left-hand limit equals the right-hand limit:
  
• 3. The limit equals the function value — .

If any one of these conditions fails, the function is discontinuous at . Geometrically, a continuous function can be drawn without lifting the pen from the paper.

We check the three conditions for continuity at .

Condition 1: is defined.

Given . ✓

Condition 2: The limit exists.

This is a standard limit that exists and equals . ✓

Condition 3: Limit equals function value.

✓

Since all three conditions are satisfied:

The function is continuous at .

If and are two functions both continuous at , then the following functions are also continuous at :

• Sum: is continuous at .
• Difference: is continuous at .
• Product: is continuous at .
• Scalar multiple: is continuous at , for any constant .
• Quotient: is continuous at , provided .
• Composition: If is continuous at and is continuous at , then is continuous at .

Consequences:

• Every polynomial function is continuous everywhere on .
• A rational function is continuous at all points where .
• Functions like , are continuous everywhere.

These rules follow directly from the algebra of limits.

For to be continuous at , we require:

Compute the limit for :

Factor the numerator using :

Substitute :

For continuity, must equal this limit:

One-One (Injective) Function:

A function is one-one if distinct elements of have distinct images in . That is:

• Example: on is one-one.

Onto (Surjective) Function:

A function is onto if every element of is the image of at least one element of . That is, range $= $ co-domain.

• Example: , is onto since every real has a cube root.

Bijective Function:

A function that is both one-one and onto. Such functions have an inverse.

• Example: , is bijective.

Key difference: Injective concerns distinctness of images; surjective concerns coverage of the co-domain; bijective combines both.

Direct substitution gives , an indeterminate form. We rationalize the numerator.

Multiply numerator and denominator by the conjugate :

The numerator becomes :

Cancel :

Substitute :

Answer:

Polynomial Function:

General form:

where are real constants and is a non-negative integer.

• Domain .
• Example: .

Rational Function:

General form:

where and are polynomials and .

• Domain .
• Example: , domain .

Exponential Function:

General form:

• Domain , Range .
• Example: or .

Continuity at :

The modulus function is:

• is defined: .
• Right-hand limit: .
• Left-hand limit: .

Since LHL RHL , the function is continuous at .

Non-differentiability (conceptual):

The derivative measures the slope of the tangent. For :

• Slope for is .
• Slope for is .

At these left and right slopes disagree, producing a sharp corner in the V-shaped graph. Because there is no unique tangent line, is not differentiable at , even though it is continuous. This shows continuity does not imply differentiability.

General Method:

For a rational function as , divide both numerator and denominator by the highest power of appearing in the denominator. Terms of the form as .

A useful shortcut based on degrees:

• If degree of numerator degree of denominator: limit .
• If degrees are equal: limit ratio of leading coefficients.
• If degree of numerator degree of denominator: limit .

Evaluation:

Divide numerator and denominator by (highest power):

As , all terms :

Answer: (equal degrees, ratio of leading coefficients).