Revision: Calculus >> Limits and Derivatives Maths Commerce (English Medium) Class 11 CBSE

“Calculus is the study of continuous (not discrete) changes in mathematical quantities.”

1. \[\lim_{x\to0}\frac{\sin x}{x}=1=\lim_{x\to0}\frac{x}{\sin x}\]

2. $$\lim_{x\to0}\frac{\tan x}{x}=1=\lim_{x\to0}\frac{x}{\tan x}$$

3. \[\lim_{x\to0}\frac{\sin^{-1}x}{x}=1=\lim_{x\to0}\frac{x}{\sin^{-1}x}\]

4. \[\lim_{x\to0}\frac{\tan^{-1}x}{x}=1=\lim_{x\to0}\frac{x}{\tan^{-1}x}\]

5. \[\lim_{x\to0}\frac{\sin x^{\circ}}{x}=\frac{\pi}{180}\]

6. \[\lim_{x\to0}\cos x=1\]

7. \[\lim_{x\to0}\frac{\sin\mathrm{k}x}{x}=\lim_{x\to0}\frac{\tan\mathrm{k}x}{x}=\mathrm{k}\]

8. \[\lim_{x\to\infty}\frac{\sin x}{x}=\lim_{x\to\infty}\frac{\cos x}{x}=0\]

9. \[\lim_{x\to\infty}\frac{\sin\left(\frac{1}{x}\right)}{\frac{1}{x}}=1=\lim_{x\to\infty}\frac{\tan\left(\frac{1}{x}\right)}{\frac{1}{x}}\]

10. \[\lim_{x\to a}\frac{\sin\left(x-a\right)}{x-a}=1=\lim_{x\to a}\frac{\tan\left(x-a\right)}{x-a}\]

Theorem :  Let f and g be two functions such that their derivatives are defined in a common domain. Then

(i) Derivative of sum of two functions is sum of the derivatives of the functions. 
`d/(dx)`[f(x) + g(x)] = `d/(dx)` f(x) + `d/(dx)`

(ii) Derivative of difference of two functions is difference of the derivatives of the functions. 
`d/(dx)` [ f(x) - g(x)] = `d/(dx)` f(x) - `d/(dx)` g(x).

(iii) Derivative of product of two functions is given by the following product rule. 
`d/(dx)`[f(x) . g(x)] = `d/(dx)` f(x) . g(x) + f(x) .`d/(dx)` g(x)

(iv) Derivative of quotient of two functions is given by the following quotient rule (whenever the denominator is non–zero).
`d/(dx)(f(x)/g(x))` =`[d/(dx) f(x) . g(x) - f(x) d/(dx) g(x)]/((g(x))^2)`

Theorem : Let f(x) = `a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0`  be a polynomial function, where `a_is`  are all real numbers and an ≠ 0. Then, the derivative function is given by
`(df(x))/(dx)` = `na_nx^(n-1) a_(n-1)x^(x-2) + ... + 2a_2x + a_1`.

Derivative of trignometric function

1. `d/(dx)` sin x = cos x 

2. `d/(dx)` cos x = - sin x

3.  `d/(dx)` tan x = `sec ^2 x `

4. `d/(dx)` sec x = sec x tan x 

5. `d/(dx)`  cosec x = - cosec x cot x

6. `d/(dx)`  cot x =` - cosec^2 x`

7. `d/(dx)`  log x = `1/ x`

8. `d/(dx)` constant = 0

9. `d/(dx) x^n = n x ^(n - 1)`

Theorem - For any positive integer n,
`lim_(x -> a) (x^n - a^n)/ (x - a)` = ` na^(n - 1)`

The expression in the above theorem for the limit is true even if n is any rational number and a is positive.

Proof :  Dividing `(x^n – a^n)` by (x – a), we see that

`x^n – a^n = (x–a) (x^(n–1) + x^(n–2) a + x^(n–3) a^2 + ... + x a^(n–2) + a^(n–1))`

Thus ,

`lim_(x-> a) (x^n - a^n)/(x - a) = lim_(x -> a) (x^(n-1) + x^(n-2) a + x^(n-3) a^2 + ... + x a^(n - 2) + a^(n - 1))`

`= a^(n – l) + a a^(n–2) +. . . + a^(n–2) (a) +a^(n–l)`

`= a^(n–1) + a^(n – 1) +...+a^(n–1) + a^(n–1)`  (n terms)

`= na ^(n - 1)` .