Limits and Derivatives

Calculus begins where elementary algebra encounters division by zero. The concept of a limit allows mathematicians to study the behavior of a function near a point, even if the function is not defined at that point itself.

The expression $x \to a$ (read as "$x$ approaches $a$") means that the variable $x$ takes values closer and closer to the real number $a$, from either the left ($x < a$) or the right ($x > a$), but $\mathbf{x \neq a}$.

Note: If $\text{LHL} \neq \text{RHL}$, the limit does not exist at $x = a$. For example, for $f(x) = \frac{|x|}{x}$ at $x = 0$, $\text{LHL} = -1$ and $\text{RHL} = +1$, so $\lim_{x \to 0} \frac{|x|}{x}$ does not exist.

When directly substituting $x = a$ into a rational function yields expressions such as $\mathbf{\frac{0}{0}}$, $\mathbf{\frac{\infty}{\infty}}$, $\infty - \infty$, or $0 \times \infty$, the form is termed indeterminate. It does NOT mean the limit does not exist; it signifies that direct substitution has failed and algebraic techniques must be deployed to determine the true limit.

Limits satisfy intuitive algebraic properties that allow complex functions to be decomposed into simpler constituent limits.

Let $f(x)$ and $g(x)$ be two real functions such that $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, where $L$ and $M$ are finite real numbers:

The Sandwich Theorem is the theoretical linchpin used to prove the fundamental trigonometric limit $\lim_{x \to 0} \frac{\sin x}{x} = 1$ by trapping $\frac{\sin x}{x}$ between $\cos x$ and $1$.

Standard limit theorems provide exact evaluation formulas without requiring tedious polynomial division or geometry for every problem.

Proof for Positive Integer $n$: Recall the algebraic factorization:

Dividing by $(x - a)$ for $x \neq a$ gives $n$ terms, each evaluating to $a^{n-1}$ as $x \to a$:

Derivation of $\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$ using Double-Angle Identity:

Crucial Examiner Note: If the angle is given in degrees (e.g., $\lim_{x \to 0} \frac{\sin x^\circ}{x}$), you MUST convert degrees to radians first: $x^\circ = \frac{\pi x}{180}$ radians, giving $\lim_{x \to 0} \frac{\sin(\pi x / 180)}{x} = \frac{\pi}{180}$.

Exponential and logarithmic limits govern continuous compound growth and appear continuously throughout calculus.

The mathematical constant $e \approx 2.71828\dots$ is defined as the limit of continuous compound interest:

Proof Sketch: Using logarithmic properties and the definition of $e$:

The derivative represents the crowning achievement of introductory calculus: quantifying instantaneous rates of change and determining exact tangents to non-linear curves.

• Physical Interpretation: If $s(t)$ is displacement at time $t$, average velocity over $[t, t + \Delta t]$ is $\frac{\Delta s}{\Delta t}$. The instantaneous velocity at time $t$ is the limit as $\Delta t \to 0$: $$v(t) = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t} = \frac{ds}{dt}$$
• Geometric Interpretation: Let $P(x, f(x))$ and $Q(x + h, f(x + h))$ be two points on the curve $y = f(x)$. The slope of the secant line passing through $P$ and $Q$ is: $$m_{\text{secant}} = \frac{f(x + h) - f(x)}{h}$$ As $h \to 0$, point $Q$ slides along the curve toward $P$, and the secant line rotates into the tangent line at $P$. Hence, the slope of the tangent line is: $$m_{\text{tangent}} = \tan \theta = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \mathbf{f'(x)}$$

1. Power Function $f(x) = x^n$: $$f'(x) = \lim_{h \to 0} \frac{(x + h)^n - x^n}{h} = \lim_{t \to x} \frac{t^n - x^n}{t - x} = \mathbf{n x^{n-1}} \quad (\text{substituting } t = x + h)$$
2. Sine Function $f(x) = \sin x$: $$f'(x) = \lim_{h \to 0} \frac{\sin(x + h) - \sin x}{h} = \lim_{h \to 0} \frac{2 \cos\left(x + \frac{h}{2}\right) \sin\left(\frac{h}{2}\right)}{h} = \lim_{h \to 0} \cos\left(x + \frac{h}{2}\right) \times \lim_{h \to 0} \frac{\sin(h/2)}{h/2} = \cos x \times 1 = \mathbf{\cos x}$$
3. Cosine Function $f(x) = \cos x$: $$f'(x) = \lim_{h \to 0} \frac{\cos(x + h) - \cos x}{h} = \lim_{h \to 0} \frac{-2 \sin\left(x + \frac{h}{2}\right) \sin\left(\frac{h}{2}\right)}{h} = -\sin x \times 1 = \mathbf{-\sin x}$$
4. Exponential Function $f(x) = e^x$: $$f'(x) = \lim_{h \to 0} \frac{e^{x + h} - e^x}{h} = e^x \lim_{h \to 0} \frac{e^h - 1}{h} = e^x \times 1 = \mathbf{e^x}$$

While the first principle provides the rigorous definition, operational differentiation rules allow rapid computation of derivatives of composite expressions.