Limits - JEE Toppers’ Notes

1. Definition and Properties of Limits

• Definition: Limit of a function $f(x)$ as $x$ approaches $a$, denoted as $\underset{x\to a}{lim}f(x)=L$, if for any given $ϵ>0$, there exists a $\delta >0$ such that $$|x-a|<\delta ⟹|f(x)-L|<ϵ$$

• 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 $\underset{x\to a}{lim}f(x)=\underset{x\to a}{lim}h(x)=L$, then $\underset{x\to a}{lim}g(x)=L$.

2. Limits at Infinity

• Definition:
• Limit at infinity: $\underset{x\to \mathrm{\infty }}{lim}f(x)=L$ if, for any $ϵ>0$, there exists $M>0$ such that for all $x>M$, we have $|f(x)-L|<ϵ$.
• Limit at negative infinity: $\underset{x\to -\mathrm{\infty }}{lim}f(x)=L$ if, for any $ϵ>0$, there exists $N<0$ such that for all $x<N$, we have $|f(x)-L|<ϵ$.

3. One-Sided Limits

• Definition:
• Right-hand limit: $\underset{x\to a^{+}}{lim}f(x)=L$ if, for every $ϵ>0$, there exists $\delta >0$ such that whenever $0<x-a<\delta$, we have $|f(x)-L|<ϵ$.
• Left-hand limit: $\underset{x\to a^{-}}{lim}f(x)=L$ if, for every $ϵ>0$, there exists $\delta >0$ such that whenever $a-\delta <x<a$, we have $|f(x)-L|<ϵ$.

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 $\underset{x\to a}{lim}f(x)=\underset{x\to a}{lim}g(x)=0$ or both approach $\pm \mathrm{\infty }$, then $$\underset{x\to a}{lim}\frac{f(x)}{g(x)}=\underset{x\to a}{lim}\frac{f^{\prime }(x)}{g^{\prime }(x)}$$ 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 $\underset{x\to a}{lim}f(x)=\underset{x\to a}{lim}h(x)=L$, then $\underset{x\to a}{lim}g(x)=L$.
• Sandwich theorem: If $f(x)\le g(x)\le h(x)$ for all $x$ in an open interval containing $a$, and if $\underset{x\to a}{lim}f(x)=\underset{x\to a}{lim}h(x)=L$, then $\underset{x\to a}{lim}g(x)=L$.

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