Notes from Toppers
Limits - JEE Toppers’ Notes
1. Definition and Properties of Limits
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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 $$|x - a| < \delta \implies |f(x) - L| < \epsilon$$
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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. One-Sided Limits
- Definition:
- Right-hand 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 $|f(x) - L| < \epsilon$.
- Left-hand 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 sum-to-product formulas, and factorize to eliminate indeterminate forms.
6. Limits Involving Logarithmic Functions
- Strategies: Rewriting using logarithmic properties, like product-to-sum and exponent-to-product 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