# Limits and Derivatives Notes, Class 11 Mathematics | SwaVid Limits and Derivatives, NCERT Class 11 Mathematics: notes with key ideas, formulas, common mistakes and 3 quick checks with worked answers. Canonical: https://www.swavid.com/cbse/class-11/maths/chapter/limits-and-derivatives Source: https://www.swavid.com/cbse/class-11/maths/chapter/limits-and-derivatives # Limits and Derivatives ## Formulas and definitions ## Key ideas ## Common mistakes ## Quick checks ## Keep going ## Questions parents ask ## Class 11 Mathematics on SwaVid ### Intuitive idea of limits ### Algebra of limits ### Limits of polynomials and rational functions ### Limits of trigonometric functions ### Derivative at a point and first principles ### Algebra of derivatives ### Next chapter ### Related chapter ### Related chapter ### All chapters SwaVid › CBSE › Class 11 › Mathematics › Limits and Derivatives Notes written by SwaVid from the NCERT textbook. Class 11 Mathematics, Chapter 12. Key ideas, formulas and common mistakes. CBSE does not print marks for a single chapter. Short answer. This chapter introduces the fundamental concepts of calculus, specifically the intuitive idea of limits and derivatives of algebraic and trigonometric functions. Students learn how to evaluate limits algebraically and compute rates of change using first principles as well as standard differentiation rules. Source: NCERT Mathematics (Reprint 2026-27) , Chapter 12, Limits and Derivatives The limit of a function f(x) as x approaches a is the value that f(x) gets closer to as x gets arbitrarily close to a from either side. A limit exists if and only if both the left hand limit and the right hand limit exist and are equal. Limits can be distributed across standard arithmetic operations. The limit of a sum, difference, product, or quotient of two functions equals the sum, difference, product, or quotient of their individual limits, provided the denominator limit is non-zero. For any polynomial function, the limit as x approaches a is simply the value of the polynomial at a. For a rational function where direct substitution results in an indeterminate form 0/0, common factors must be cancelled out before evaluating the limit. Trigonometric limits are evaluated using standard theorems such as the sandwich theorem. Two fundamental standard limits are that sin(x)/x approaches 1 and (1 - cos(x))/x approaches 0 as x approaches 0, where x is in radians. The derivative of a function f(x) at a point represents the instantaneous rate of change of the function with respect to x. Finding the derivative using the limit of the difference quotient [f(x + h) - f(x)] / h as h approaches 0 is called differentiation from first principles. Derivatives follow linearity rules along with the product rule and quotient rule. The derivative of a product u·v is u'v + uv', and the derivative of a quotient u/v is (u'v - uv') / v². Try each question first. Then tap it to see the answer. Substitute x = 2 to see the form: (2² - 4) / (2 - 2) = 0/0, which is indeterminate. Factorise the numerator: x² - 4 = (x - 2)(x + 2). Cancel the common term (x - 2) since x ≠ 2: (x - 2)(x + 2) / (x - 2) = x + 2. Evaluate the remaining expression: lim (x → 2) (x + 2) = 2 + 2 = 4. 4 Rewrite the expression in the form of standard limit sin(θ)/θ: (1/2) · [sin(4x) / x]. Multiply and divide the denominator by 4: (1/2) · 4 · [sin(4x) / (4x)] = 2 · [sin(4x) / (4x)]. As x → 0, 4x → 0, so lim (4x → 0) sin(4x) / (4x) = 1. Compute the final product: 2 · 1 = 2. 2 Identify u = x + 1 and v = x - 1, which gives u' = 1 and v' = 1. Apply the quotient rule: f'(x) = [u'v - uv'] / v². Substitute the components: f'(x) = [1 · (x - 1) - (x + 1) · 1] / (x - 1)². Simplify the numerator: (x - 1 - x - 1) / (x - 1)² = -2 / (x - 1)². -2 / (x - 1)² Statistics . Relations and Functions . Trigonometric Functions . The 14 chapters of Class 11 Mathematics . The CBSE Class 11 Maths curriculum says it gives no chapter or unit-wise weightage. Theory is 80 marks, plus 20 for internal assessment. No. They are written by SwaVid from the NCERT textbook. The CBSE curriculum is the reference for what is assessed. Statistics. SwaVid is an AI tutor for Indian students in Classes 6 to 12. It finds the earlier ideas a child missed, then teaches today’s chapter. Start free to see how it works. - Ch 12 NCERT book - 80 marks, theory paper - 3 quick checks - Directly substituting x = a when it results in 0/0 . Do not write 0/0 as 0 or undefined. Factorise the expressions, rationalise, or apply standard limit identities before evaluating. - Reversing the terms in the numerator of the quotient rule . Remember that the quotient rule requires differentiating the numerator first: (u'v - uv') / v², not (uv' - u'v) / v². - Using degree values in standard trigonometric limits . Standard formulas like lim (x → 0) sin(x) / x = 1 hold only when x is in radians. Convert degrees to radians by multiplying by π / 180 first. - Forgetting the negative sign when differentiating cosine . Recall that the derivative of cos(x) is -sin(x), whereas the derivative of sin(x) is +cos(x). ## Key Links - [CBSE](https://www.swavid.com/cbse) - [Class 11](https://www.swavid.com/cbse/class-11) - [Mathematics](https://www.swavid.com/cbse/class-11/maths/chapters) - [Start free →](https://www.swavid.com/signup) - [All 14 chapters](https://www.swavid.com/cbse/class-11/maths/chapters) - [NCERT Mathematics (Reprint 2026-27)](https://ncert.nic.in/textbook.php) - [Open notes](https://www.swavid.com/cbse/class-11/maths/chapter/statistics) - [Open notes](https://www.swavid.com/cbse/class-11/maths/chapter/relations-and-functions) - [Open notes](https://www.swavid.com/cbse/class-11/maths/chapter/trigonometric-functions) - [Class 11 Mathematics](https://www.swavid.com/cbse/class-11/maths/chapters) - [Start free →](https://www.swavid.com/signup)