Notes from Toppers
Limits  JEE Toppers’ Notes
1. Definition and Properties of Limits

Definition: Limit of a function $f(x)$ as $x$ approaches $a$, denoted as $\lim\limits_{x \to a} f(x) = L$, if for any given $\epsilon > 0$, there exists a $\delta > 0$ such that $$x  a < \delta \implies f(x)  L < \epsilon$$

Properties:
 Limit laws: These involve properties like sum, difference, product, and quotient of limits, as well as laws for constant multiples and compositions.
 Squeeze theorem: If $f(x) \le g(x) \le h(x)$ for all $x$ in an open interval containing $a$, except possibly at $a$ itself, and if $\lim\limits_{x \to a} f(x) = \lim\limits_{x \to a} h(x) = L$, then $\lim\limits_{x \to a} g(x) = L$.
2. Limits at Infinity
 Definition:
 Limit at infinity: $\lim\limits_{x \to \infty} f(x) = L$ if, for any $\epsilon > 0$, there exists $M > 0$ such that for all $x > M$, we have $f(x)  L < \epsilon$.
 Limit at negative infinity: $\lim\limits_{x \to \infty} f(x) = L$ if, for any $\epsilon > 0$, there exists $N < 0$ such that for all $x < N$, we have $f(x)  L < \epsilon$.
3. OneSided Limits
 Definition:
 Righthand limit: $\lim\limits_{x \to a^+} f(x) = L$ if, for every $\epsilon > 0$, there exists $\delta > 0$ such that whenever $0 < x  a < \delta$, we have $f(x)  L < \epsilon$.
 Lefthand limit: $\lim\limits_{x \to a^} f(x) = L$ if, for every $\epsilon > 0$, there exists $\delta > 0$ such that whenever $a  \delta < x < a$, we have $f(x)  L < \epsilon$.
4. Continuity
 Definition: A function $f(x)$ is said to be continuous at a point $c$ if
 (f(c)) is defined
 (\lim\limits_{x \to c} f(x) = f(c))
5. Limits Involving Trigonometric Functions
 Strategies: Convert trigonometric expressions into algebraic expressions using trigonometric identities, simplify expressions using sumtoproduct formulas, and factorize to eliminate indeterminate forms.
6. Limits Involving Logarithmic Functions
 Strategies: Rewriting using logarithmic properties, like producttosum and exponenttoproduct transformations, as well as applying natural logarithmic derivatives to handle indeterminate forms.
7. Limits Involving Exponential Functions
 Strategies: Rewrite exponential expressions using exponent properties to eliminate indeterminate forms.
8. L’Hôpital’s Rule
 Definition: If $\lim\limits_{x \to a} f(x) = \lim\limits_{x \to a} g(x) = 0$ or both approach $\pm \infty$, then $$\lim\limits_{x \to a} \frac{f(x)}{g(x)} = \lim\limits_{x \to a} \frac{f’(x)}{g’(x)}$$ provided the limit on the right side exists or is infinite.
9. Squeeze Theorem and Related Theorems
 Squeeze theorem If $f(x) \le g(x) \le h(x)$ for all $x$ in an open interval containing $a$, and $\lim\limits_{x \to a} f(x) = \lim\limits_{x \to a} h(x) = L$, then $\lim\limits_{x \to a} g(x) = L$.
 Sandwich theorem: If $f(x) \le g(x) \le h(x)$ for all $x$ in an open interval containing $a$, and if $\lim\limits_{x \to a} f(x) = \lim\limits_{x \to a} h(x) = L$, then $\lim\limits_{x \to a} g(x) = L$.
10. Applications of Limits
 Finding limits can help determine derivatives, evaluate integrals, plot graphs, study asymptotic behavior, determine convergence or divergence of series, and identify points of discontinuity or undefined behavior.
Referred NCERT Books:
 “NCERT Mathematics,” Class 11, by R.D. Sharma
 “NCERT Mathematics,” Class 12, by Amit M. Agarwal