Understanding Limit, Continuity, and Differentiability in Calculus

Limit, continuity, and differentiability are fundamental concepts in calculus dealing with the local and global behaviour of real-valued functions. The limit describes the value a function approaches as the input nears a specific point; continuity characterises functions with no abrupt jumps or breaks; differentiability establishes the existence of a unique tangent or instantaneous rate of change at each point where the function is defined.

Formal Definition of Limit, Left-Hand and Right-Hand Limits for Real-Valued Functions

A function $f(x)$ is said to have limit $L$ as $x \to a$ if, for every $\varepsilon > 0$, there exists $\delta > 0$ such that $|f(x) - L| < \varepsilon$ whenever $0 < |x - a| < \delta$. This is notated as $\displaystyle \lim_{x \to a} f(x) = L$.

The left-hand limit of $f(x)$ at $x = a$ exists if $\displaystyle \lim_{x \to a^-} f(x)$ exists, defined as the value approached by $f(x)$ as $x$ approaches $a$ from values less than $a$.

The right-hand limit of $f(x)$ at $x = a$ exists if $\displaystyle \lim_{x \to a^+} f(x)$ exists, defined as the value approached by $f(x)$ as $x$ approaches $a$ from values greater than $a$.

The two-sided limit at $x = a$ exists if and only if both the left-hand and right-hand limits exist and are equal at $x = a$.

Algebraic Properties and Standard Results for Limits

The following algebraic properties hold for limits: If $\displaystyle \lim_{x \to a} f(x) = l_1$ and $\displaystyle \lim_{x \to a} g(x) = l_2$, then $\displaystyle \lim_{x \to a} [f(x) + g(x)] = l_1 + l_2$ and $\displaystyle \lim_{x \to a} [f(x) \cdot g(x)] = l_1 \cdot l_2$.

If $l_2 \neq 0$, then $\displaystyle \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{l_1}{l_2}$.

If $c$ is a constant, then $\displaystyle \lim_{x \to a} c = c$ and $\displaystyle \lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)$.

Standard limits include: $\displaystyle \lim_{x \to 0} \frac{\sin x}{x} = 1$, $\displaystyle \lim_{x \to 0} \frac{\tan x}{x} = 1$, $\displaystyle \lim_{x \to 0} \frac{e^x - 1}{x} = 1$, $\displaystyle \lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1$, $\displaystyle \lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}$, $\displaystyle \lim_{x \to \infty} \left(1 + \frac{k}{x}\right)^{mx} = e^{mk}$.

The Sandwich (Squeeze) Theorem states: If for $x$ near $a$, $h(x) \leq f(x) \leq g(x)$ and $\displaystyle \lim_{x \to a} h(x) = \lim_{x \to a} g(x) = l$, then $\displaystyle \lim_{x \to a} f(x) = l$.

Limits Solved Examples can help to practice the application of these properties.

Indeterminate Forms and L'Hospital's Rule

If substitution into a limit results in the indeterminate forms $\frac{0}{0}$ or $\frac{\infty}{\infty}$, L'Hospital's Rule allows the computation by evaluating $\displaystyle \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$, provided the new limit exists.

Rationalisation and factorisation techniques are frequently required when evaluating limits involving roots or polynomials leading to indeterminate forms.

Types of Discontinuity in Real Functions

Removable discontinuity occurs when the left-hand and right-hand limits at $x = a$ both exist, are finite and equal, but are not equal to $f(a)$ or $f(a)$ is undefined.

Jump discontinuity occurs if both one-sided limits at $x = a$ exist and are finite, but are unequal.

Infinite discontinuity occurs if at least one of the one-sided limits at $x = a$ is infinite.

Criterion for Continuity at a Point

A function $f(x)$ is continuous at $x = a$ if and only if $\displaystyle \lim_{x \to a} f(x)$ exists and equals $f(a)$. Explicitly, $\displaystyle \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$.

Continuity over an interval requires continuity at each interior point and appropriate one-sided continuity at the endpoints.

Algebra of Continuous Functions and Theorems

If $f(x)$ and $g(x)$ are both continuous at $x = a$, then $f(x) + g(x)$, $f(x)g(x)$, $kf(x)$ for any constant $k$, and $\frac{f(x)}{g(x)}$ (provided $g(a) \neq 0$) are all continuous at $x = a$.

A composition of continuous functions is continuous; if $g(x)$ is continuous at $x = a$ and $f(y)$ is continuous at $y = g(a)$, then $f(g(x))$ is continuous at $x = a$.

The Intermediate Value Theorem states that if a function is continuous on a closed interval $[a, b]$, then it takes every value between $f(a)$ and $f(b)$ at some point in $[a, b]$.

The Extreme Value Theorem guarantees that a continuous function on $[a, b]$ attains both minimum and maximum values within $[a, b]$.

Definition and Algebraic Criterion for Differentiability

A function $f(x)$ is said to be differentiable at $x = a$ if the following limit exists and is finite: $\displaystyle \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} = f'(a)$.

Existence of the derivative at $a$ implies continuity at $a$, but continuity does not necessarily imply differentiability.

The function is differentiable on an interval if the above limit exists for every point in that interval.

If a function is defined piecewise, differentiability at the point of joining requires both continuity and matching left-hand and right-hand derivatives.

Standard Derivatives and Theorems Associated with Differentiability

Standard derivatives include $\frac{d}{dx}(x^n) = n x^{n-1}$, $\frac{d}{dx}(\sin x) = \cos x$, $\frac{d}{dx}(\cos x) = -\sin x$, $\frac{d}{dx}(\tan x) = \sec^2 x$, $\frac{d}{dx}(\ln x) = \frac{1}{x}$, $\frac{d}{dx}(e^{x}) = e^x$, as well as corresponding formulas for inverse trigonometric and exponential functions.

The Chain Rule states that the derivative of a composite function $f(g(x))$ is $f'(g(x)) \cdot g'(x)$.

Function compositions, implicit functions, and parametric forms require specific methods for differentiation, including the parametric derivative $\displaystyle \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{\frac{d}{dt} \left(\frac{dy}{dx}\right)} {\frac{dx}{dt}}$ if $x$ and $y$ are both functions of $t$.

Further discussion on differentiability of composite functions is available at the Differentiability Of Composite Functions page.

Mean Value Theorems and Their Criteria

Rolle’s Theorem states: If $f(x)$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c \in (a, b)$ such that $f'(c) = 0$.

Lagrange’s Mean Value Theorem (LMVT): If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists at least one $c \in (a, b)$ such that $\displaystyle f'(c) = \frac{f(b) - f(a)}{b - a}$.

Example – Limit Using Expansion and L'Hospital’s Rule

Evaluate $\displaystyle \lim_{x \to 0} \frac{e^x - 1 - x}{x^2}$

Given: $\displaystyle \lim_{x \to 0} \frac{e^x - 1 - x}{x^2}$

Expand $e^x$ using the Taylor series: $e^x = 1 + x + \frac{x^2}{2} + \cdots$

Substitute: $e^x - 1 - x = \frac{x^2}{2} + o(x^2)$

Thus, $\displaystyle \frac{e^x - 1 - x}{x^2} = \frac{\frac{x^2}{2} + o(x^2)}{x^2} = \frac{1}{2} + o(1)$

Taking the limit as $x \to 0$, the result is $\frac{1}{2}$.

Example – Continuity and Differentiability for Piecewise Function

Let $f(x) = \begin{cases} ax + b & x < 1 \\ x^2 & x \geq 1 \end{cases}$. Determine $a$ and $b$ for $f(x)$ to be continuous and differentiable at $x = 1$.

For continuity at $x = 1$: $\displaystyle \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1)$

$\lim_{x \to 1^-} f(x) = a \cdot 1 + b = a + b$

$\lim_{x \to 1^+} f(x) = (1)^2 = 1$

Set $a + b = 1$

For differentiability, set left and right derivatives equal at $x = 1$:

Left derivative: $f'_-(1) = a$

Right derivative: $f'_+(1) = \frac{d}{dx}(x^2)_{x=1} = 2$

Set $a = 2$

Therefore $b = 1 - a = -1$

The parameters $a = 2$, $b = -1$ ensure the function is both continuous and differentiable at $x = 1$.

For more in-depth practice, refer to the Limit Continuity And Differentiability Practice Paper.

Key Observations and Common Exam Cues

Every differentiable function is continuous at the point of differentiability, but not every continuous function is necessarily differentiable.

Discontinuity or non-differentiability often occurs at points of piecewise definition, greatest integer, modulus, or cusp points, as in $f(x) = |x|$ at $x = 0$.

Success in problems involving limits of complex functions may require repeated use of expansions, L'Hospital's Rule, or identification of indeterminate forms beyond simple substitution.

Explicit checking of both continuity and derivative match at boundary points is mandatory for piecewise and greatest integer-related functions in exam settings.

Thorough familiarity with expansion, algebraic manipulation, and precise application of definitions is required for full marks in JEE Main and Advanced assessments.