By going through these CBSE Class 11 Maths Notes Chapter 13 Limits and Derivatives Class 11 Notes, students can recall all the concepts quickly.

## Limits and Derivatives Notes Class 11 Maths Chapter 13

Left Hand Limit: The $$\lim _{x \rightarrow a^{-}}$$f(x) is the expected value f(x) at x = a, given the values of fix) near x to the left of a. This value is called the left hand limit of f(x) at a.

Right Hand Limit : The $$\lim _{x \rightarrow a^{+}}$$f(x) is the expected value of f(x) at x – a, given the values of fix) near x to the right of a. ‘ ‘his value is called the right hand limit of f(x) at a.

Limit of a Function: If the right and left hand limits coincide, the common value of the limit of f(x) as x → a is called the limit of a function. It is denoted by $$\lim _{x \rightarrow a}$$f(x).

Algebra of Limits:
(i) Limit of sum of two functions is sum of the limits of the functions, i.e.,
$$\lim _{x \rightarrow a}$$ [(x)+g(x)] = $$\lim _{x \rightarrow a}$$f(x) + $$\lim _{x \rightarrow a}$$ g(x).

(ii) Limit of difference of two functions is difference of the limits of the functions, i.e.,
$$\lim _{x \rightarrow a}$$ [(x)-g(x)] = $$\lim _{x \rightarrow a}$$f(x) – $$\lim _{x \rightarrow a}$$ g(x).

(iii) Limit of product of two functions is product of the limits
of the functions i.e.,
$$\lim _{x \rightarrow a}$$ [(x)+g(x)] . $$\lim _{x \rightarrow a}$$f(x) . $$\lim _{x \rightarrow a}$$ g(x).

(iv) Limit of quotient of two functions is quotient of the limits of the functions (whenever the denominator is non-zero i.e..
$$\lim _{c \rightarrow a} \frac{f(x)}{g(x)}=\frac{\lim _{x \rightarrow a} f(x)}{\lim _{x \rightarrow a} g(x)}$$

(v) $$\lim _{x \rightarrow a}$$ (c.f)(x) = c$$\lim _{x \rightarrow a}$$f(x)

Limit of Polynomial:
Let f(x) = a0 + a1x + a + a2x2 + … + anxn be a polynomial function.
Let $$\lim _{x \rightarrow a}$$ xk = ak.
$$\lim _{x \rightarrow a}$$ f(x) = [a0 + a1x + a + a2x2 + … + anxn]
= a0 + a1$$\lim _{x \rightarrow a}$$x + a + a2$$\lim _{x \rightarrow a}$$x2 + … + an$$\lim _{x \rightarrow a}$$xn
= a0 + x1a + a2a2 + … + anan
= f(a).

Limit of Rational Function : A function f is said to be a rational function, if f(x) = $$\frac{g(x)}{h(x)}$$, where g(x) and h(x) are polynomials h(x) ≠ 0

However, ifg(a) = 0 and h(a) = 0, i.e., this is of the form 0/0 then factor(s),x-a of g(x) and h(x) are determined and then cancelled out.
Let g(x) = (x-a)p(x)
h(x) = (x-a)q(x)

(ii) For any positive integer n,
$$\lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}$$ = nan-1

Theorems:
(i) Let f and g be two real valued functions with the same domain such that f(x) ≤ g(x) for allx in the domain of definition. For some a, if both $$\lim _{x \rightarrow a}$$f(x) and $$\lim _{x \rightarrow a} g(x) exist, then [latex]\lim _{x \rightarrow a}f(x) ≤ [latex]\lim _{x \rightarrow a}$$g(x)

(ii) Sandwich Theorem : Let f, g and h be real functions
such that f(x) ≤ g(x) ≤ h(x) for all x in the common domain of
definition. For some real number a, if $$\lim _{x \rightarrow a}$$g(x) f(x) = Z, $$\lim _{x \rightarrow a}$$g(x) = h(x) = l, then lim g(x) = l.

Limit of Trigonometric Functions :
(i) $$\lim _{x \rightarrow 0} \frac{\sin x}{x}$$ = 1
(ii) $$\lim _{x \rightarrow 0} \frac{1-\cos x}{x}$$ = 0

Derivative of f(x) at x = a : Suppose f is a real valued of function and a is a point in its domain of definition. The derivative of f at x = a is defined by
$$\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$$
provided this limit exists. It is denoted by f'(a).

Derivative of fix) : Suppose f is a real valued function, the function defined by $$\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ wherever the limit exists is defined to be the derivative of/‘and is denoted by fix). Thus,

f ‘(x) = $$\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$

Algebra of Derivative of Functions:
(i) Derivative of sum of two functions is sum of the derivatives of functions:
$$\frac{d}{d x}$$[f(x) + g(x)] = [$$\frac{d}{d x}$$f(x) + $$\frac{d}{d x}$$g(x)]
or (u + v)’ = u’ + v’.

iii) Derivative of difference of two functions is the difference of the derivatives of the functions :
$$\frac{d}{d x}$$[f(x) – g(x)] = [$$\frac{d}{d x}$$f(x) – $$\frac{d}{d x}$$g(x)]
or (u + v)’ = u’ + v’.

(iii) Derivative of product of two functions is given by the product rule, i.e.,
or (u .v)’ = u’v + uv’.

(iv) Derivative of quotient of two functions is given by quotient rule (whenever the denominator is non-zero).

(v) Derivative of λf(x)
$$\frac{d}{d x}$$[λf(x)] = λ $$\frac{d y}{d x}$$f(x)
or (λu)’ = λu’

Some Derivatives:
(i) $$\frac{d}{d x}$$xn = nxn-1
(ii) $$\frac{d}{d x}$$(ax + b)n = na(ax + b)n-1
(iii) If f(x) = a0xn + a1xn-1 + a2xn-2 + … am
then na0xn-1 + (n-1)a1xn-2 + (n-2)a2xn-3+ …….. +an-1
(iv) $$\frac{d}{d x}$$ (sin x) = cos x.
(v) $$\frac{d}{d x}$$ (cos x) = – sin x.
(vi) $$\frac{d}{d x}$$ (tan x) = sec2 x.
(vii) $$\frac{d}{d x}$$ (cosec x) = – cosec x cot x.
(viii) $$\frac{d}{d x}$$ (sec x) = sec x tan x.
(ix) $$\frac{d}{d x}$$ (cot x) = cosec2 x