Chapter 12: Limits and Derivatives

This chapter is your first taste of calculus , the mathematics of change. Calculus answers two related questions:

1. How does a function behave near a point? (limits)
2. How fast is it changing at that point? (derivatives)

A limit captures what value a function $f(x)$ "approaches" as $x$ approaches some target $a$, even when $f(a)$ itself is undefined or behaves oddly. We write $\lim_{x \to a} f(x) = L$ to mean "as $x$ gets close to $a$, $f(x)$ gets close to $L$". Limits underlie everything in calculus.

A derivative is a rate of change. The derivative $f'(a)$ measures how fast $f(x)$ is changing at the instant $x = a$. Geometrically, $f'(a)$ is the slope of the tangent line to the graph $y = f(x)$ at $(a, f(a))$. Physically, if $f(t)$ is the position of a particle at time $t$, then $f'(t)$ is its instantaneous velocity.

The chapter introduces both concepts informally (no $\varepsilon$-$\delta$ formalism), provides the basic algebra of limits, presents standard limits like $\lim_{x \to 0} \dfrac{\sin x}{x} = 1$, and then defines the derivative as a special limit: $f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}.$ From there we derive the standard differentiation rules: linearity, product rule, quotient rule, derivatives of polynomials, sine, cosine.

For Class XII, JEE, and beyond, calculus becomes the central topic. The two ideas in this chapter , limit and derivative , are the seeds from which all of differential and integral calculus grows.

What's inside

1. Informal limits and the idea of approach.
2. Algebra of limits, evaluating simple limits.
3. Standard limits including $\sin x / x$, $(a^x - 1)/x$, $(1 + x)^{1/x}$.
4. Derivative from first principles.
5. Differentiation rules , linearity, product, quotient.
6. Derivatives of standard functions , polynomials, trig, exponential.

Key results / Formula card

How to read this chapter

Don't try to prove limits with $\varepsilon$-$\delta$ yet. Build intuition first. After each example, draw a quick sketch of the function near the point of interest. Memorise the table of standard limits , they appear constantly. Practice "by first principles" until you can derive $(x^2)' = 2x$ and $(\sin x)' = \cos x$ in your sleep.