JEE Limits Cheatsheet

Cheatsheet Content

### Introduction to Limits - **Definition:** A function $f(x)$ has a limit $L$ as $x$ approaches $a$, written as $\lim_{x \to a} f(x) = L$, if for every $\epsilon > 0$, there exists a $\delta > 0$ such that $|f(x) - L| ### Fundamental Theorems on Limits If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$: 1. **Sum Rule:** $\lim_{x \to a} [f(x) \pm g(x)] = L \pm M$ 2. **Product Rule:** $\lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M$ 3. **Quotient Rule:** $\lim_{x \to a} \left[\frac{f(x)}{g(x)}\right] = \frac{L}{M}$, provided $M \neq 0$ 4. **Scalar Multiple Rule:** $\lim_{x \to a} [c \cdot f(x)] = c \cdot L$ 5. **Power Rule:** $\lim_{x \to a} [f(x)]^n = L^n$ 6. **Constant Function:** $\lim_{x \to a} c = c$ 7. **Identity Function:** $\lim_{x \to a} x = a$ ### Indeterminate Forms When direct substitution of $x=a$ into $\lim_{x \to a} f(x)$ results in one of these forms, further evaluation techniques are required: - $\frac{0}{0}$ - $\frac{\infty}{\infty}$ - $0 \times \infty$ - $\infty - \infty$ - $0^0$ - $\infty^0$ - $1^\infty$ ### Evaluation Techniques #### 1. Factorization and Cancellation - Used for $\frac{0}{0}$ forms, especially with polynomial or rational functions. - Factor the numerator and denominator to cancel common terms $(x-a)$. #### 2. Rationalization - Used when square root terms are present in $\frac{0}{0}$ forms. - Multiply numerator and denominator by the conjugate to eliminate square roots. #### 3. L'Hôpital's Rule - Applicable for $\frac{0}{0}$ or $\frac{\infty}{\infty}$ forms. - If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$, provided the latter limit exists. Repeat if necessary. #### 4. Using Standard Limits - **Algebraic:** - $\lim_{x \to a} \frac{x^n - a^n}{x - a} = na^{n-1}$ - **Trigonometric:** - $\lim_{x \to 0} \frac{\sin x}{x} = 1$ - $\lim_{x \to 0} \frac{\tan x}{x} = 1$ - $\lim_{x \to 0} \frac{\sin^{-1} x}{x} = 1$ - $\lim_{x \to 0} \frac{\tan^{-1} x}{x} = 1$ - $\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$ - $\lim_{x \to 0} \frac{\sec x - 1}{x^2} = \frac{1}{2}$ - **Exponential and Logarithmic:** - $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$ - $\lim_{x \to 0} \frac{a^x - 1}{x} = \ln a$ $(a > 0)$ - $\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$ - $\lim_{x \to 0} (1+x)^{1/x} = e$ - $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$ - $\lim_{x \to a} [f(x)]^{g(x)}$ of $1^\infty$ form: $e^{\lim_{x \to a} [f(x)-1]g(x)}$ #### 5. Using Series Expansions - For simple functions around $x=0$: - $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$ - $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$ - $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$ - $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots$ (for $|x| ### Limits at Infinity - For rational functions $\lim_{x \to \infty} \frac{P(x)}{Q(x)}$: - If $\text{deg}(P) = \text{deg}(Q)$: limit is ratio of leading coefficients. - If $\text{deg}(P) \text{deg}(Q)$: limit is $\pm \infty$. - Divide numerator and denominator by the highest power of $x$ in the denominator. ### Important Notes - **Continuity:** If $f(x)$ is continuous at $x=a$, then $\lim_{x \to a} f(x) = f(a)$. - **Greatest Integer Function:** $\lim_{x \to a} [x]$ (where $[.]$ is GIF) - Does not exist if $a$ is an integer. - Exists if $a$ is not an integer. - **Fractional Part Function:** $x - [x]$ - Does not exist at integers.