Limit and Its Fundamentals

Concept Explanation:

The limit of a function describes the value that the function approaches as the input approaches a particular point. For a function \( f(x) \), the limit as \( x \) approaches \( a \) is denoted by

\[ \lim_{x \to a} f(x) = L \]

if \( f(x) \) gets arbitrarily close to \( L \) as \( x \) approaches \( a \) (but \( x \neq a \)).

The left-hand limit (LHL) and right-hand limit (RHL) are defined as:

• Left-hand limit: \( \lim_{x \to a^-} f(x) = L_1 \), the value approached as \( x \) approaches \( a \) from the left.
• Right-hand limit: \( \lim_{x \to a^+} f(x) = L_2 \), the value approached as \( x \) approaches \( a \) from the right.

The limit \( \lim_{x \to a} f(x) \) exists if and only if \( L_1 = L_2 = L \).

Formula Derivation

For polynomial functions \( f(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \), the limit is

\[ \lim_{x \to a} f(x) = a_0 + a_1 a + a_2 a^2 + \cdots + a_n a^n = f(a) \]

For rational functions \( f(x) = \frac{g(x)}{h(x)} \), where \( g(x) \) and \( h(x) \) are polynomials and \( h(a) \neq 0 \),

\[ \lim_{x \to a} f(x) = \frac{g(a)}{h(a)} \]

If \( g(a) = 0 \) and \( h(a) = 0 \), factorization and cancellation are used to evaluate the limit.

Worked Illustrations

Example: Find \( \lim_{x \to 1} (x-1)^2 \).

Left-hand limit:

Set \( x = 1 - h \), \( h \to 0^+ \), then

\[ \lim_{h \to 0^+} (1 - h - 1)^2 = \lim_{h \to 0^+} (-h)^2 = 0 \]

Right-hand limit:

Set \( x = 1 + h \), \( h \to 0^+ \), then

\[ \lim_{h \to 0^+} (1 + h - 1)^2 = \lim_{h \to 0^+} h^2 = 0 \]

Since LHL = RHL = 0, the limit exists and equals 0.

Solved Examples

Example 1: Evaluate \( \lim_{x \to 0} \frac{\sin x}{x} \).

Solution:

Using the standard limit,

\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]

Example 2: Find \( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \).

Solution:

Factor numerator:

\[ x^2 - 4 = (x - 2)(x + 2) \]

Cancel \( x - 2 \):

\[ \lim_{x \to 2} \frac{(x - 2)(x + 2)}{x - 2} = \lim_{x \to 2} (x + 2) = 4 \]

Practice Set

• Level 1 – Easy
• Find \( \lim_{x \to 3} (2x + 1) \).
• Evaluate \( \lim_{x \to 0} \frac{\sin 3x}{x} \).
• Level 2 – Moderate
• Calculate \( \lim_{x \to 1} \frac{x^3 - 1}{x - 1} \).
• Find \( \lim_{x \to 0} \frac{1 - \cos x}{x^2} \).
• Level 3 – Challenging
• Evaluate \( \lim_{x \to 0} \frac{e^x - 1}{x} \).
• Find \( \lim_{x \to 1} \frac{\sqrt{x} - 1}{x - 1} \).

Answer Key

• Level 1
• \( \lim_{x \to 3} (2x + 1) = 2(3) + 1 = 7 \)
• \( \lim_{x \to 0} \frac{\sin 3x}{x} = 3 \) (using \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \))
• Level 2
• \( \lim_{x \to 1} \frac{x^3 - 1}{x - 1} = \lim_{x \to 1} (x^2 + x + 1) = 3 \)
• \( \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2} \)
• Level 3
• \( \lim_{x \to 0} \frac{e^x - 1}{x} = 1 \)
• Rationalize numerator:

\[ \lim_{x \to 1} \frac{\sqrt{x} - 1}{x - 1} = \lim_{x \to 1} \frac{(\sqrt{x} - 1)(\sqrt{x} + 1)}{(x - 1)(\sqrt{x} + 1)} = \lim_{x \to 1} \frac{x - 1}{(x - 1)(\sqrt{x} + 1)} = \lim_{x \to 1} \frac{1}{\sqrt{x} + 1} = \frac{1}{2} \]

Quick Reference

Concept	Formula
Limit of polynomial	\( \lim_{x \to a} f(x) = f(a) \)
Limit of rational function	\( \lim_{x \to a} \frac{g(x)}{h(x)} = \frac{g(a)}{h(a)} \), if \( h(a) \neq 0 \)
Standard limit	\( \lim_{x \to 0} \frac{\sin x}{x} = 1 \)
Left-hand limit	\( \lim_{x \to a^-} f(x) \)
Right-hand limit	\( \lim_{x \to a^+} f(x) \)

Glossary

• Limit: The value a function approaches as the input approaches a point.
• Left-hand limit: Limit as input approaches from the left side.
• Right-hand limit: Limit as input approaches from the right side.
• Indeterminate form: Expressions like \( \frac{0}{0} \) that require algebraic manipulation to evaluate limits.
• Rational function: A function expressed as the ratio of two polynomials.

Derivatives

Concept Explanation:

The derivative of a function \( f(x) \) at a point \( a \) measures the instantaneous rate of change of the function at that point. It is defined as the limit of the difference quotient:

\[ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \]

provided this limit exists.

Formula Derivation

Starting from the definition, the derivative at \( a \) is the slope of the tangent line to the curve \( y = f(x) \) at \( x = a \).

For a polynomial \( f(x) = a_n x^n + \cdots + a_1 x + a_0 \), the derivative is

\[ f'(x) = n a_n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \cdots + a_1 \]

Worked Illustrations

Example: Find the derivative of \( f(x) = \frac{1}{x} \) using the definition.

\[ f'(x) = \lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h} = \lim_{h \to 0} \frac{\frac{x - (x+h)}{x(x+h)}}{h} = \lim_{h \to 0} \frac{-h}{h x (x+h)} = \lim_{h \to 0} \frac{-1}{x (x+h)} = -\frac{1}{x^2} \]

Solved Examples

Example 1: Find \( \frac{d}{dx} (x^3 + 2x) \).

Solution:

Using power rule:

\[ \frac{d}{dx} x^3 = 3x^2, \quad \frac{d}{dx} 2x = 2 \]

Therefore,

\[ \frac{d}{dx} (x^3 + 2x) = 3x^2 + 2 \]

Example 2: Differentiate \( f(x) = \sin x \).

Solution:

\[ \frac{d}{dx} (\sin x) = \cos x \]

Practice Set

• Level 1 – Easy
• Find \( \frac{d}{dx} (5x^2) \).
• Differentiate \( f(x) = 3x + 7 \).
• Level 2 – Moderate
• Find \( \frac{d}{dx} (x^3 - 4x) \).
• Differentiate \( f(x) = \cos x \).
• Level 3 – Challenging
• Find \( \frac{d}{dx} \left( \frac{1}{x^2} \right) \) using the definition.
• Differentiate \( f(x) = x^n \) where \( n \) is any real number.

Answer Key

• Level 1
• \( \frac{d}{dx} (5x^2) = 10x \)
• \( \frac{d}{dx} (3x + 7) = 3 \)
• Level 2
• \( \frac{d}{dx} (x^3 - 4x) = 3x^2 - 4 \)
• \( \frac{d}{dx} (\cos x) = -\sin x \)
• Level 3
• Using definition,

\[ \frac{d}{dx} \left( \frac{1}{x^2} \right) = \lim_{h \to 0} \frac{\frac{1}{(x+h)^2} - \frac{1}{x^2}}{h} = \lim_{h \to 0} \frac{x^2 - (x+h)^2}{h x^2 (x+h)^2} = \lim_{h \to 0} \frac{-2xh - h^2}{h x^2 (x+h)^2} = \lim_{h \to 0} \frac{-2x - h}{x^2 (x+h)^2} = -\frac{2}{x^3} \]

• \( \frac{d}{dx} (x^n) = n x^{n-1} \)

Quick Reference

Derivative Rule	Formula
Power Rule	\( \frac{d}{dx} (x^n) = n x^{n-1} \)
Sum Rule	\( (u + v)' = u' + v' \)
Product Rule	\( (uv)' = u'v + uv' \)
Quotient Rule	\( \left( \frac{u}{v} \right)' = \frac{v u' - u v'}{v^2} \)
Derivative of \( \sin x \)	\( \cos x \)
Derivative of \( \cos x \)	\( -\sin x \)

Glossary

• Derivative: Instantaneous rate of change of a function at a point.
• Difference quotient: \( \frac{f(a+h) - f(a)}{h} \), average rate of change over interval \( h \).
• Power rule: Derivative of \( x^n \) is \( n x^{n-1} \).
• Product rule: Derivative of product \( uv \) is \( u'v + uv' \).
• Quotient rule: Derivative of quotient \( \frac{u}{v} \) is \( \frac{v u' - u v'}{v^2} \).