Limits and Derivatives

Sir Issac Newton, based on his concepts of rate and change, set forth the basic laws of differential calculus and integral calculus came forth as the reverse process. Calculus and the basics of differentiation and calculus serve as the foundation for advanced mathematics, modern physics and various other branches of modern sciences and engineering. Limits and derivatives serve as the entry point to calculus as a topic of class 11 maths for CBSE students.

Limits

In Mathematics, a limit is defined as a value that a function approaches as the input approaches some value. Limits are important in calculus and mathematical analysis and used to define integrals, derivatives, and continuity.

To express the limit of a function, we represent it as:

\(\displaystyle {\lim _{n\to c}f(n)=L}\)

Properties of Limits

Let p and q be two functions and a be a value such that \(\displaystyle{\lim_{x \to a}p(x)}\) and \(\displaystyle{\lim_{x \to a}q(x)}\) exists.

1.\(\displaystyle{\lim_{x \to a}[p(x) + g (x)] = \lim_{x \to a}p(x) + \lim_{x \to a}g (x)}\)

2.\(\displaystyle{\lim_{x \to a}[p(x) – g (x)] = \lim_{x \to a}p(x) – \lim_{x \to a}g (x)}\)

3.For every real number k,

\(\displaystyle{\lim_{x \to a}[k p(x)] = k \lim_{x \to a}p(x)}\)

4.\(\displaystyle{\lim_{x \to a}[p(x)\; q(x)] = \lim_{x \to a}p(x) \times \lim_{x \to a}q(x)}\)

5.\(\displaystyle{\lim_{x \to a}\frac{p(x)}{q(x)}} =\frac{\displaystyle{\lim_{x \to a}p(x)}}{\displaystyle{\lim_{x \to a}q(x)}}\)

For more detailed information about limits and sample problems, check out our page – Limits.

Derivatives

Whereas a derivative refers to the instantaneous rate of change of a quantity with respect to the other. It helps to investigate the moment by moment nature of an amount. The Derivative of a function is represented as:

\(\displaystyle{\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}}\)

For the function f, its derivative is said to be f'(x) given the equation above exists.

Properties of Derivatives

Since the very definition of derivatives involves limits in a rather direct fashion, we expect the rules of derivatives to follow closely that of limits as given below:

1.\(\frac{d}{dx}\;[p(x)+q(x)]= \frac{d}{dx}\;(p(x))+ \frac{d}{dx}\;(q(x))\)

2.\(\frac{d}{dx}\;[p(x)-q(x)]= \frac{d}{dx}\;(p(x))- \frac{d}{dx}\;(q(x))\)

3.\(\frac{d}{dx}\;[p(x)\times q(x)]= \frac{d}{dx}\;[p(x)]\;q(x) + p(x)\;\frac{d}{dx}\;[q(x)]\)

4.\(\frac{d}{dx}\left [\frac{p(x)}{q(x)} \right ]= \frac{\frac{d}{dx}\;[p(x)]\;q(x) – p(x)\;\frac{d}{dx}\;[q(x)] }{(g(x))^2}\)<

For more detailed information about derivatives and sample problems, check out our page – derivatives.


Practise This Question

The number of proper subsets of the set {1, 2, 3} is