Class 11 Maths Limits and Derivatives Notes

Limits and Derivatives – Class 11 Maths (NCERT Based)

The chapter Limits and Derivatives is the first step into calculus for Class 11 students. It introduces the concept of limits, continuity of functions, and derivatives, which are used to study rate of change, slope of curves, and maxima/minima problems.

This guide is NCERT-aligned, student-friendly, and suitable for exam preparation.

---

📖 1. Limits of a Function

The limit of a function f(x) as x approaches a value a is the value that f(x) approaches:$$\lim_{x \to a} f(x) = L$$x→alim​f(x)=L

Key Properties:

1. $\lim_{x \to a} [f(x) ± g(x)] = \lim f(x) ± \lim g(x)$limx→a​[f(x)±g(x)]=limf(x)±limg(x)
2. $\lim_{x \to a} [f(x) × g(x)] = \lim f(x) × \lim g(x)$limx→a​[f(x)×g(x)]=limf(x)×limg(x)
3. $\lim_{x \to a} [f(x)/g(x)] = \lim f(x)/\lim g(x), \text{ if } g(a) \neq 0$limx→a​[f(x)/g(x)]=limf(x)/limg(x), if g(a)=0

---

🔹 Standard Limits

1. $\lim_{x \to 0} \frac{\sin x}{x} = 1$limx→0​xsinx​=1
2. $\lim_{x \to 0} \frac{1 – \cos x}{x} = 0$limx→0​x1−cosx​=0
3. $\lim_{x \to 0} (1 + x)^{1/x} = e$limx→0​(1+x)1/x=e

---

📖 2. Continuity

A function f(x) is continuous at x = a if:

1. f(a) is defined
2. $\lim_{x \to a} f(x)$limx→a​f(x) exists
3. $\lim_{x \to a} f(x) = f(a)$limx→a​f(x)=f(a)

A function continuous at every point of its domain is called continuous function.

---

📖 3. Derivatives of a Function

The derivative of a function f(x) at a point x measures the rate of change of the function:$$f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$$f′(x)=h→0lim​hf(x+h)−f(x)​

• Derivative represents the slope of the tangent at a point on the curve y = f(x).

---

🔹 Basic Derivative Formulas

Function	Derivative
$x^n$xn	$nx^{n-1}$nxn−1
$\sin x$sinx	$\cos x$cosx
$\cos x$cosx	$-\sin x$−sinx
$e^x$ex	$e^x$ex
$\ln x$lnx	$1/x$1/x

---

🔹 Rules of Differentiation

1. Sum Rule: $(f ± g)’ = f’ ± g’$(f±g)′=f′±g′
2. Product Rule: $(fg)’ = f’g + fg’$(fg)′=f′g+fg′
3. Quotient Rule: $(f/g)’ = \frac{f’g − fg’}{g^2}$(f/g)′=g2f′g−fg′​
4. Chain Rule: If y = f(u) and u = g(x), then $\frac{dy}{dx} = \frac{dy}{du} × \frac{du}{dx}$dxdy​=dudy​×dxdu​