# 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 - 3

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