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Mathematics · Class 11 Science

Ch 7Limits and Derivatives — Class 11 Mathematics, concept-first.

Before limits can be defined precisely, it helps to build the idea of a limit informally -- by asking what value a function gets closer and closer to as itself gets closer and closer to some fixed number , without ever actually reaching .

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Key concepts

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Derivative as Slope of the Tangent

For a curve and a fixed point on it, a nearby second point determines a secant line through and whose slope is the difference quotient -- the same expression that defines the derivative algebraically.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

1

Intuitive Idea of Limit

Before limits can be defined precisely, it helps to build the idea of a limit informally -- by asking what value a function gets closer and closer to as itself gets closer and closer to some fixed num…

2

Algebra of Limits

Evaluating a limit directly from a table of values, as in Section 1, is instructive but impractical for anything beyond the simplest functions.

3

Limits of Polynomial and Rational Functions

Limits of polynomials. A polynomial is built entirely out of the constant function and the identity function , combined only by addition, subtraction, scalar multiplication and (repeated) multiplicati…

4

Limits of Trigonometric Functions

Direct substitution for sine and cosine. The functions and are defined for every real number (the unit-circle definition covered earlier) and their graphs are unbroken, continuous curves with no jumps…

5

Limits of Exponential and Logarithmic Functions

Direct substitution. The exponential function (base ) is defined and continuous for every real , so by direct substitution, exactly as for polynomials and trigonometric functions.

6

Derivative as Rate of Change

Average velocity. Suppose a particle moves along a straight line, and its position (the distance covered, measured from a fixed starting point) at time is given by a function .

7

Derivative as Slope of the Tangent

From secant to tangent. Let be a curve, and fix a point on it. Choose a second, nearby point on the same curve, for some small .

8

Definition of Derivative

Formal definition. Let be a function defined in some open interval containing the point . The derivative of at , written , is defined as provided this limit exists.

9

Algebra of Derivatives

Exactly as the algebra of limits (Section 2) allows the limit of a combination of functions to be built from the limits of the simpler functions inside it, the algebra of derivatives allows the deriva…

10

Derivatives of Polynomial and Trigonometric Functions

With the algebra of derivatives (Section 9) in hand, only the derivatives of the basic building blocks -- powers of , and , -- are still needed to differentiate any polynomial or trigonometric express…

Summary

Limits. exists iff the left-hand and right-hand limits both equal . Algebra of limits: limits of a sum/difference/product/quotient (denominator limit ) equal the sum/difference/product/quotient of the…

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 6 questions6 questions
  1. Q1The value of lim(x→4) (e^x - e^4)/(x-4) is (a) e^-4 (b) e^4 (c) 1 (d) None of these.Preview
  2. Q2If f(x) = |x|, then f'(0) is (a) 0 (b) 1 (c) -1 (d) None of these.Preview
  3. Q3Evaluate: lim(x→π/6) (√3 sin x - cos x)/(x - π/6).Preview
  4. Q4Prove that the derivative of an odd function is an even function.Preview
  5. Q5If 2f(x)+f(-x) = 1+x, find f'(10), where f'(x) denotes derivative of f(x).Preview
  6. Q6Evaluate lim(x→π/4) (4√2-(cos x+sin x)^5)/(1-sin 2x).Preview

More questions

34 Q
+Show 3 questions3 questions
  1. Q32Find the equation of the tangent to the curve $y=x^2$ at the point $(2,4)$.Free
  2. Q33Find the derivative of $y=x\sin x+\cos x$.Preview
  3. Q34A particle moves along a straight line so that the distance covered in $t$ seconds is $s(t)=4t^2-3t+5$ metres. Find (a) the average velocity…Preview
+Show 10 questions10 questions
  1. Example 1If $\lim_{x\to a}f(x)=5$ and $\lim_{x\to a}g(x)=-2$, find the value of $\lim_{x\to a}\big[3f(x)-2g(x)\big]$.Free
  2. Example 2Evaluate $\lim_{x\to1}\big(2x^3-x^2+4x-1\big)$.Free
  3. Example 3Evaluate $\lim_{x\to2}\dfrac{x^2-4}{x-2}$.Free
  4. Example 4Evaluate $\lim_{x\to1}\dfrac{x^3-1}{x-1}$ using the standard result $\lim_{x\to a}\dfrac{x^n-a^n}{x-a}=na^{n-1}$.Preview
  5. Example 5Evaluate $\lim_{x\to0}\dfrac{\sin4x}{x}$.Preview
  6. Example 6Evaluate $\lim_{x\to0}\dfrac{1-\cos x}{x^2}$.Preview
  7. Example 7Evaluate $\lim_{x\to0}\dfrac{e^{3x}-1}{x}$.Preview
  8. Example 8Find the derivative of $f(x)=x^2+3x$ from first principles.Preview
  9. Example 9If $u(x)$ and $v(x)$ are differentiable functions with $u(3)=4,\ u'(3)=2,\ v(3)=5,\ v'(3)=-3$, find the value of $(uv)'(3)$ and $\left(\dfra…Preview
  10. Example 10Differentiate $y=x^4-\sin x+\cos x$.Preview
+Show 3 questions3 questions
  1. Q11By considering values of $x$ close to $3$, on both sides, determine $\lim_{x\to3}\dfrac{x^2-9}{x-3}$.Free
  2. Q12Find $\lim_{x\to0^-}\dfrac{|x|}{x}$ and $\lim_{x\to0^+}\dfrac{|x|}{x}$. Does $\lim_{x\to0}\dfrac{|x|}{x}$ exist?Preview
  3. Q13If $f(x)=\begin{cases}x+1, & x<2\\ 3x-1, & x\ge2\end{cases}$, find $\lim_{x\to2}f(x)$ if it exists.Preview
+Show 3 questions3 questions
  1. Q14Evaluate $\lim_{x\to2}\big(3x^3-2x^2+x-5\big)$.Free
  2. Q15Evaluate $\lim_{x\to2}\dfrac{x^5-32}{x-2}$ using the standard limit formula.Preview
  3. Q16Evaluate $\lim_{x\to-1}\dfrac{x^3+1}{x+1}$.Preview
+Show 3 questions3 questions
  1. Q17Evaluate $\lim_{x\to0}\dfrac{\tan5x}{x}$.Free
  2. Q18Evaluate $\lim_{x\to0}\dfrac{\sin3x}{\sin5x}$.Preview
  3. Q19Evaluate $\lim_{x\to\pi/2}\dfrac{1-\sin x}{\cos^2x}$.Preview
+Show 3 questions3 questions
  1. Q20Evaluate $\lim_{x\to0}\dfrac{e^{5x}-1}{x}$.Free
  2. Q21Evaluate $\lim_{x\to0}\dfrac{e^x-e^{-x}}{x}$.Preview
  3. Q22Evaluate $\lim_{x\to0}\dfrac{\ln(1+3x)}{\sin2x}$.Preview
+Show 3 questions3 questions
  1. Q23Find the derivative of $f(x)=5x^2-3x+7$ from first principles.Free
  2. Q24Find the derivative of $f(x)=\dfrac{1}{x}$ ($x\neq0$) from first principles.Preview
  3. Q25Find the derivative of $f(x)=\sqrt{x}$ ($x>0$) from first principles.Preview
+Show 3 questions3 questions
  1. Q26If $f(2)=4,\ f'(2)=3,\ g(2)=-1,\ g'(2)=2$, find $(f+g)'(2)$ and $(f-g)'(2)$.Free
  2. Q27If $f(1)=2,\ f'(1)=-3,\ g(1)=5,\ g'(1)=4$, find $(fg)'(1)$ using the product rule.Preview
  3. Q28If $f(0)=6,\ f'(0)=2,\ g(0)=3,\ g'(0)=-1$ (with $g(0)\neq0$), find $\left(\dfrac{f}{g}\right)'(0)$ using the quotient rule.Preview
+Show 3 questions3 questions
  1. Q29Differentiate $y=4x^5-3x^3+2x-9$.Free
  2. Q30Differentiate $y=x^2\cos x$ using the product rule.Preview
  3. Q31Differentiate $y=\dfrac{\sin x}{x}$ ($x\neq0$) using the quotient rule.Preview