Limits – Complete JEE Main 2026 Revision

A limit describes the value a function approaches as the variable gets close to a point. It does not require the function to be defined at that point.

1. Definition of Limit

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

This means $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$.

2. Left & Right Hand Limits

$\displaystyle \lim_{x \to a^-} f(x)$ (Left Hand Limit)

$\displaystyle \lim_{x \to a^+} f(x)$ (Right Hand Limit)

Limit exists iff:

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

JEE Main frequently tests piecewise functions using this.

3. Direct Substitution Rule

If $f(x)$ is continuous at $x=a$:

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

Fails when substitution gives indeterminate form.

4. Standard Limits (Core JEE Main)

Trigonometric

$\displaystyle \lim_{x \to 0} \frac{\sin x}{x} = 1$

$\displaystyle \lim_{x \to 0} \frac{\tan x}{x} = 1$

$\displaystyle \lim_{x \to 0} \frac{1-\cos x}{x^2} = \frac12$

$\displaystyle \lim_{x \to 0} \frac{\sin ax}{bx} = \frac{a}{b}$

Note: $x$ must be in radians.

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Exponential & Logarithmic

$\displaystyle \lim_{x \to 0} \frac{e^x-1}{x} = 1$

$\displaystyle \lim_{x \to 0} \frac{a^x-1}{x} = \ln a$

$\displaystyle \lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$

$\displaystyle \lim_{x \to \infty} \left(1+\frac{1}{x}\right)^x = e$

5. Algebraic Limits

$\displaystyle \lim_{x \to a} \frac{x^n-a^n}{x-a} = na^{n-1}$

$\displaystyle \lim_{x \to 0} \frac{(1+x)^n - 1}{x} = n$

Used in JEE Main simplification-based questions.

6. Limits Involving Infinity

Rational Functions

If degree(numerator) = degree(denominator):

Limit = ratio of leading coefficients

If degree(numerator) < degree(denominator):

Limit = 0

If degree(numerator) > degree(denominator):

Limit = $\infty$ or $-\infty$

7. Indeterminate Forms

Form	Resolution
$\frac{0}{0}$	Factorization / Rationalization
$\frac{\infty}{\infty}$	Divide by highest power
$0\cdot\infty$	Convert to fraction
$\infty-\infty$	Combine terms
$0^0,1^\infty,\infty^0$	Logarithmic method

8. Sandwich (Squeeze) Theorem

If $f(x)\le g(x)\le h(x)$ and

$\lim f(x) = \lim h(x) = L$

Then:

$\displaystyle \lim g(x) = L$

Used in oscillatory functions like $x\sin\frac{1}{x}$.

9. Absolute Value Limits

$\displaystyle \lim_{x\to0} \frac{|x|}{x}$ does not exist

Break absolute values into piecewise form before evaluating.

10. JEE Main Traps

• Using degree instead of radians
• Skipping one-sided limits in piecewise functions
• Not identifying dominant term at infinity
• Blind use of L’Hospital (not in JEE Main syllabus focus)

✔ Fully aligned with JEE Main 2026 Mathematics syllabus
✔ Covers all PYQ patterns
✔ Ready bridge to Continuity & Differentiability