## 13. Limits and Derivatives # LIMITS AND DERIVATIVES CLASS 11 MATHS

## Summary

The expected value of the function as dictated by the points to the left of a point defines the left hand limit of the function at that point. Similarly the right hand limit.

Limit of a function at a point is the common value of the left and right hand limits, if they coincide.

For a function f and a real number a, $$\lim_{x \to a} {f(x)}$$ and f(a) may not be same (In fact, one may be defined and not the other one).

For functions f and g the following holds:
$$\lim_{x\to a} [f(x) \pm g(x)] = \lim_{x \to a}f(x) \pm \lim_{x \to a}g(x)$$
$$\lim_{x\to a} [f(x) \times g(x)] = \lim_{x \to a}f(x) \times \lim_{x \to a}g(x)$$
$$\lim_{x\to a} \frac{f(x)} {g(x)} = \frac {\lim_{x \to a} f(x)} {\lim_{x \to a} g(x)}$$

Following are some of the standard limits
$$\lim_{x\to a} \frac {x^n-a^n}{x-a} = n a^{n-1}$$
$$\lim_{x\to 0} \frac {sin(x)}{x} = 1$$
$$\lim_{x\to 0} \frac {1-cos(x)}{x} = 0$$

The derivative of a function f at a is defined by
$$f'(a) = \lim_{h\to 0} \frac {f(a+h)-f(a)}{h}$$

Derivative of a function f at any point x is defined by
$$f'(x) = \frac{d}{dx} f(x) = \lim_{h\to 0} \frac {f(x+h)-f(x)}{h}$$

For functions u and v the following holds:
$$(u \pm v)' = u' \pm v'$$
$$(uv)' = uv' + vu'$$
$$\left(\frac{u}{v} \right)'=\frac{u'v-uv'}{v^2}$$ provided all are defined.

Following are some of the standard derivatives.
$$\frac{d}{dx} (x^n) = n x^{n-1}$$
$$\frac{d}{dx} (sin x) = cos x$$
$$\frac{d}{dx} (cos x) = -sin x$$

## Lecture - 1

• Meaning of Limits
• Different methods of finding limits

## Lecture - 2

• NCERT Example 1, 2
• NCERT Exercise 13.1
• Identities in limits

## Lecture - 3

• Logarithmic Functions
• Exponential Functions
• NCERT Supplementary Exercise
• Few complex question

## Lecture - 4

• Limits on Double definition functions
• NCERT Exercise 13.1

## Lecture - 5

• Meaning of Derivatives
• First Principle Derivation

## Lecture - 6

• First Principle questions
• Base of Derivatives

## Lecture - 7

• NCERT Example 19, 20
• NCERT Exercise 13.2 Q4
• Miscellaneous Exercise Q1

## Lecture - 8

• Direct Derivative
• Product Rule
• Quotient Rule

## Lecture - 9

• Direct Derivatives
• NCERT Exercise 13.2

## Lecture - 10

• NCERT Miscellaneous Exercise Q2 to Q11, Q15 to Q18, Q20, Q22 to Q29

## Lecture - 11

• Chain Rule
• NCERT Miscellaneous Exercise