Introduction to Limits

Hello JEE aspirants! Welcome to the fascinating world of Limits. This topic is not just a fundamental concept in calculus but also a cornerstone for many advanced mathematical concepts you'll encounter in JEE Main. Mastering limits will give you a significant edge in tackling problems related to continuity, differentiability, and integration. So, let's dive in!

What is a Limit?

Imagine you're walking towards a destination. A limit is like describing where you're heading, even if you never actually reach the exact spot. Mathematically, the limit of a function $f(x)$ as $x$ approaches a value $a$ is the value that $f(x)$ gets closer and closer to as $x$ gets closer and closer to $a$ .

We write this as: $\lim_{x \rightarrow a} f(x) = L$

This means "the limit of $f(x)$ as $x$ approaches $a$ is equal to $L$ ."

Graphical Interpretation

Graphically, this means that as you trace the function $f(x)$ with your finger, getting closer to the $x$ -value of $a$ , the $y$ -value on the graph gets closer to $L$ .

For instance, consider the function $f(x) = x^2$ . As $x$ approaches 2, $f(x)$ approaches 4. Even though $f(2)$ is indeed 4, the limit describes the function's behavior around $x = 2$ , not necessarily at $x = 2$ . Think of a hole in the graph at $x = a$ ; the limit still exists if the function approaches a specific value from both sides.

Left-Hand Limit (LHL)

The Left-Hand Limit is the value that $f(x)$ approaches as $x$ approaches $a$ from the left side (values less than $a$ ). We denote this as:

$\lim_{x \rightarrow a^-} f(x) = L$

Imagine walking towards your destination but only allowed to approach it from the west.

For example, if you have a piecewise function like: $f(x) = \begin{cases} x + 1, & \text{if } x < 2 \\ x^2, & \text{if } x \geq 2 \end{cases}$ Then, the LHL as $x$ approaches 2 is $2+1 = 3$ because you are using the first piece of the function ( $x < 2$ ).

Right-Hand Limit (RHL)

The Right-Hand Limit is the value that $f(x)$ approaches as $x$ approaches $a$ from the right side (values greater than $a$ ). We denote this as:

$\lim_{x \rightarrow a^+} f(x) = L$

Think of approaching your destination only from the east.

Using the same piecewise function above, the RHL as $x$ approaches 2 is $2^2 = 4$ because you are using the second piece of the function ( $x \geq 2$ ).

Existence of a Limit

A limit exists at a point if and only if the Left-Hand Limit and the Right-Hand Limit are equal. This is a crucial concept!

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

If the LHL and RHL are not equal, the limit does not exist at that point. Our piecewise function above demonstrates this perfectly. Since the LHL is 3 and the RHL is 4 at $x=2$ , the limit does not exist at $x=2$ .

Important Formulas

Epsilon-Delta Definition:
$\lim_{x \rightarrow a} f(x) = L$ means for every $ε > 0$ , there exists a $δ > 0$ such that $|f(x) - L| < ε$ whenever $0 < |x - a| < δ$ .

Explanation: This formal definition states that no matter how small an interval you choose around $L$ (given by $ε$ ), you can always find an interval around $a$ (given by $δ$ ) such that every $x$ in the interval around $a$ maps to an $f(x)$ within the interval around $L$ . In simpler terms, you can make $f(x)$ as close to $L$ as you want by making $x$ close enough to $a$ .
Existence Condition:
Limit exists if and only if: $\lim_{x \rightarrow a^-} f(x) = \lim_{x \rightarrow a^+} f(x)$

Explanation: For a limit to exist at a point, the function must approach the same value from both the left and the right sides. If these two values are different, the limit does not exist.

Tips for Solving Problems

Always check for LHL and RHL: Especially for piecewise functions or functions with absolute values, checking the left and right-hand limits is crucial.
Simplify expressions: Before directly substituting the value of $x$ , try to simplify the function. Factorization, rationalization, and trigonometric identities can be very helpful.
Look for indeterminate forms: If direct substitution leads to forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ , use L'Hôpital's Rule (after confirming its applicability).

Common Mistakes to Avoid

Assuming $f(a)$ exists: The limit $\lim_{x \rightarrow a} f(x)$ does not depend on whether $f(a)$ exists or not. It only depends on the function's behavior around $a$ .
Ignoring LHL and RHL: For many functions, especially those defined piecewise, the LHL and RHL must be checked separately.
Incorrectly applying L'Hôpital's Rule: Ensure that you have an indeterminate form ( $\frac{0}{0}$ or $\frac{\infty}{\infty}$ ) before applying L'Hôpital's Rule.
Forgetting to simplify: Always try to simplify the function before evaluating the limit.

Limits are a foundational concept, and understanding them well will significantly benefit you in your JEE Main preparation. Keep practicing, and you'll master this topic in no time!