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. 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