Functions and Limits Notes for Class 12 & IIT JEE

Functions and limits are the entrance to calculus, and these Functions and Limits Notes are written for IIT JEE. In the NCERT books, functions are in Class 11 (Chapter 2, Relations and Functions) and Class 12 (Chapter 1, Relations and Functions), limits are in Class 11 (Chapter 12, Limits and Derivatives; Chapter 13 in older books), and continuity is in Class 12 (Chapter 5, Continuity and Differentiability). IIT JEE joins them in one unit. These Functions and Limits Notes follow that order, so you can use them with any of the books.

Functions and Limits Notes: Chapter Overview and Syllabus

Check the latest NTA (JEE Main) syllabus and the JEE Advanced syllabus for the exact lists. The weightage of the unit is not published, so use previous years’ papers to judge how often it is asked.

Tip from the eSaral Maths team: Do every limit problem in the same order: substitute first, then simplify, then use a standard limit, then a series, and use L’Hôpital’s rule last. Most mistakes come from using the rule before checking the form.

Functions and Limits Notes 

India's Best Exam Preparation for Class 12th - Download Now

India's Best Exam Preparation for Class 12th - Download Now

India's Best Exam Preparation for Class 12th - Download Now

India's Best Exam Preparation for Class 12th - Download Now

India's Best Exam Preparation for Class 12th - Download Now

Functions for IIT JEE in Functions and Limits Notes

A function $f : A \to B$ gives exactly one output in $B$ for every input in $A$. The domain is the set of inputs, and the range is the set of outputs. For the full basics, with the domain rules and the standard functions, see eSaral’s Functions Class 11 notes.

Special functions: the greatest integer function $[x]$, the fractional part $\{x\} = x - [x]$, the modulus $\lvert x\rvert$ and the signum function. They are the main source of limit and continuity problems, because they jump or change form at some points.

Graph Transformations

Limit of a Function: Meaning and Existence in Functions and Limits Notes

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

means that $f(x)$ gets as close as we like to $L$ when $x$ gets close to $a$ (but $x \neq a$). The value $f(a)$ does not matter, and $f(a)$ may not even be defined.

• Left-hand limit (LHL): $\displaystyle\lim_{x \to a^-} f(x)$, with $x < a$. Right-hand limit (RHL): $\displaystyle\lim_{x \to a^+} f(x)$, with $x > a$.
• The limit exists if and only if the LHL and the RHL both exist, are finite and are equal.
• For $[x]$ and $\lvert x\rvert/x$, and for any function that changes its rule at $a$, always check both sides.

Algebra of limits: if $\lim f = L$ and $\lim g = M$, then:

$$\lim (f \pm g) = L \pm M, \qquad \lim (fg) = LM, \qquad \lim \frac{f}{g} = \frac{L}{M}\ (M \neq 0), \qquad \lim \big(c\,f\big) = cL$$

Sandwich (squeeze) theorem: if $g(x) \leq f(x) \leq h(x)$ near $a$, and $\lim g = \lim h = L$, then $\lim f = L$. For example, $\displaystyle\lim_{x \to 0} x^{2}\sin\frac{1}{x} = 0$.

Standard Limits for IIT JEE in Functions and Limits Notes

$$\lim_{x \to 0}\frac{\sin x}{x} = 1, \quad \lim_{x \to 0}\frac{\tan x}{x} = 1, \quad \lim_{x \to 0}\frac{\sin^{-1} x}{x} = 1, \quad \lim_{x \to 0}\frac{\tan^{-1} x}{x} = 1$$

(the angle is in radians)

Methods to Evaluate Limits in Functions and Limits Notes

A limit of the form $\dfrac{0}{0}$, $\dfrac{\infty}{\infty}$, $0 \cdot \infty$, $\infty - \infty$, $1^{\infty}$, $0^{0}$ or $\infty^{0}$ is an indeterminate form, and it needs work.

L’Hôpital’s rule: if $f$ and $g$ are differentiable, and $\dfrac{f(x)}{g(x)}$ is of the form $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ at $x = a$, then

$$\lim_{x \to a}\frac{f(x)}{g(x)} = \lim_{x \to a}\frac{f'(x)}{g'(x)}$$

provided the limit on the right exists. You can apply it again if the new form is still indeterminate. Other forms must first be changed into $0/0$ or $\infty/\infty$: write $0 \cdot \infty$ as $\dfrac{0}{1/\infty}$, and combine $\infty - \infty$ into one fraction.

The form $1^{\infty}$: if $f(x) \to 1$ and $g(x) \to \infty$ as $x \to a$, then

$$\lim_{x \to a}\big[f(x)\big]^{g(x)} = e^{\lim_{x \to a} g(x)\,[\,f(x) - 1\,]}$$

Series expansions (for $x$ near $0$):

$$\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \cdots, \qquad \cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \cdots, \qquad \tan x = x + \frac{x^{3}}{3} + \cdots$$ $$e^{x} = 1 + x + \frac{x^{2}}{2!} + \cdots, \qquad \ln(1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \cdots, \qquad (1 + x)^{n} = 1 + nx + \frac{n(n-1)}{2!}x^{2} + \cdots$$

Limits at Infinity in Functions and Limits Notes

For a rational function $\dfrac{P(x)}{Q(x)}$ with the degrees $p$ and $q$, as $x \to \infty$:

For an expression with a square root, such as $\sqrt{x^{2} + x} - x$, multiply and divide by the conjugate. A horizontal asymptote $y = L$ comes from $\displaystyle\lim_{x \to \pm\infty} f(x) = L$, and a vertical asymptote $x = a$ comes from an infinite limit at $a$.

Continuity and Discontinuity in Functions and Limits Notes

A function $f$ is continuous at $x = a$ if

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

Useful facts:

• Sums, differences, products and composites of continuous functions are continuous, and a quotient is continuous where the denominator is not zero.
• Polynomials, $\sin x$, $\cos x$, $e^{x}$ and $\lvert x\rvert$ are continuous everywhere. $[x]$ is discontinuous at every integer.
• Intermediate value property: if $f$ is continuous on $[a, b]$, it takes every value between $f(a)$ and $f(b)$. If $f(a)\,f(b) < 0$, then $f(x) = 0$ has a root in $(a, b)$.
• A function that is differentiable at a point is continuous there, but a continuous function need not be differentiable (for example, $\lvert x\rvert$ at $0$).

Key Formulas in Functions and Limits Notes

Solved Examples from Functions and Limits Notes

Example 1 (domain): Find the domain of $f(x) = \sqrt{x^{2} - 5x + 6} + \dfrac{1}{\sqrt{4 - x}}$.

We need $x^{2} - 5x + 6 = (x - 2)(x - 3) \geq 0$, so $x \leq 2$ or $x \geq 3$. We also need $4 - x > 0$, so $x < 4$. The domain is $(-\infty, 2] \cup [3, 4)$.

Example 2 (inverse function): Find the inverse of $f(x) = \dfrac{2x + 1}{x - 3}$.

Put $y = \dfrac{2x + 1}{x - 3}$. Then $y(x - 3) = 2x + 1$, so $x(y - 2) = 3y + 1$ and $x = \dfrac{3y + 1}{y - 2}$. Hence $f^{-1}(x) = \dfrac{3x + 1}{x - 2}$, with $x \neq 2$.

Example 3 (period): Find the period of $f(x) = \lvert\sin x\rvert + \lvert\cos x\rvert$.

Each term has the period $\pi$, but $f\left(x + \dfrac{\pi}{2}\right) = \lvert\cos x\rvert + \lvert\sin x\rvert = f(x)$, so the period is $\dfrac{\pi}{2}$.

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

$\dfrac{(x - 2)(x + 2)}{x - 2} = x + 2$, so the limit is $4$.

Example 5 (standard limit): Find $\displaystyle\lim_{x \to 0}\frac{\sin 5x}{3x}$.

$\dfrac{\sin 5x}{3x} = \dfrac{5}{3}\cdot\dfrac{\sin 5x}{5x} \to \dfrac{5}{3}$.

Example 6 (power limit): Find $\displaystyle\lim_{x \to 1}\frac{x^{5} - 1}{x^{3} - 1}$.

$\dfrac{x^{5} - 1}{x - 1} \to 5$ and $\dfrac{x^{3} - 1}{x - 1} \to 3$, so the limit is $\dfrac{5}{3}$.

Example 7 (limit at infinity): Find $\displaystyle\lim_{x \to \infty}\left(\sqrt{x^{2} + x} - x\right)$.

Multiply and divide by the conjugate: $\dfrac{x}{\sqrt{x^{2} + x} + x} = \dfrac{1}{\sqrt{1 + 1/x} + 1} \to \dfrac{1}{2}$.

Example 8 (the form $1^{\infty}$): Find $\displaystyle\lim_{x \to 0}(1 + 2x)^{1/x}$.

Here $f - 1 = 2x$ and $g = \dfrac{1}{x}$, so $\lim g(f - 1) = 2$ and the limit is $e^{2}$.

Example 9 (series method): Find $\displaystyle\lim_{x \to 0}\frac{\sin x - x}{x^{3}}$.

$\sin x - x = -\dfrac{x^{3}}{6} + \cdots$, so the limit is $-\dfrac{1}{6}$.

Example 10 (L’Hôpital’s rule): Find $\displaystyle\lim_{x \to 0}\frac{e^{x} - 1 - x}{x^{2}}$.

The form is $\dfrac{0}{0}$. Differentiate: $\dfrac{e^{x} - 1}{2x}$ is again $\dfrac{0}{0}$. Differentiate again: $\dfrac{e^{x}}{2} \to \dfrac{1}{2}$.

Example 11 (the form $0 \cdot \infty$): Find $\displaystyle\lim_{x \to 0^+} x\ln x$.

Write it as $\dfrac{\ln x}{1/x}$, which is $\dfrac{\infty}{\infty}$. Then $\dfrac{1/x}{-1/x^{2}} = -x \to 0$.

Example 12 (left and right limits): Does $\displaystyle\lim_{x \to 2}[x]$ exist?

LHL $= 1$ and RHL $= 2$. They are not equal, so the limit does not exist, and $[x]$ has a jump discontinuity at $x = 2$.

Example 13 (finding a constant for continuity): Find $k$ so that $f(x) = \dfrac{\sin 3x}{x}$ for $x \neq 0$ and $f(0) = k$ is continuous at $0$.

$\displaystyle\lim_{x \to 0}\frac{\sin 3x}{x} = 3$, so $k = 3$.

Example 14 (a trigonometric limit): Find $\displaystyle\lim_{x \to 0}\frac{\tan x - \sin x}{x^{3}}$.

$\tan x - \sin x = \tan x\,(1 - \cos x) \approx x \cdot \dfrac{x^{2}}{2}$, so the limit is $\dfrac{1}{2}$.

Example 15 (the number $e$): Find $\displaystyle\lim_{x \to \infty}\left(1 + \frac{3}{x}\right)^{x}$.

$\lim g(f - 1) = \lim x \cdot \dfrac{3}{x} = 3$, so the limit is $e^{3}$.

Common Mistakes in Functions and Limits Notes Problems

How to Study Functions and Limits Notes

1. Learn the functions first from eSaral’s Functions Class 11 notes, and download the NCERT books from eSaral’s NCERT books for Class 11 page (limits are in Class 11) and the NCERT books for Class 12 page (continuity is in Class 12).
2. Memorise the standard limits and the series, and practise the order of methods: substitute, simplify, standard limit, series, and L’Hôpital’s rule.
3. Practise one problem each on the forms $\dfrac{0}{0}$, $\dfrac{\infty}{\infty}$, $1^{\infty}$, $0\cdot\infty$ and $\infty - \infty$, then on limits with $[x]$ and $\lvert x\rvert$, and then on continuity with a constant.
4. Test yourself with eSaral’s JEE Main chapter-wise PYQs and JEE Advanced chapter-wise PYQs.