Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 14 Limit and Differentiation will enable students to study smartly.

## CHSE Odisha 11th Class Math Notes Chapter 14 Limit and Differentiation

**Limit Of A Function:**

A real number ‘l’ is called the limit of the function f(x) as x tends to ‘a’ if for every ∈ > 0, there exist δ > 0 such that |f(x) – l| < ∈ whenever |x – a| < δ

We write \(\lim _{x \rightarrow a}\) f(x) = l

Left and right hand limit:

Left hand limit of f(x) as x → a is:

\(\lim _{x \rightarrow a-}\) f(x) = \(\lim _{x \rightarrow 0}\) f(a – h)

Right hand limit of f(x) as x → a is:

\(\lim _{x \rightarrow a+}\) f(x) = \(\lim _{h \rightarrow 0}\) f(a + h)

Existance of limit:

\(\lim _{x \rightarrow a}\) f(x) exists if it is unique, irrespective of any type of approach i.e if LHL = RHL. i.e if \(\lim _{x \rightarrow a-}\) f(x) = \(\lim _{x \rightarrow a+}\) f(x)

Indeterminate forms:

The forms : \(\frac{0}{0}, \frac{\infty}{\infty}\), ∞ – ∞, 0 × ∞, 0°, ∞° and 1^{∞} are called indeterminate forms in mathematics.

Properties of limit:

Some standard limits:

**Limit At Infinite And Infinite Limits:**

(a) We write \(\lim _{x \rightarrow a}\) f(x) = ∞ if for a given m > 0, there exists δ > 0 such that |x – a| < δ ⇒ f(x) > m for large m.

(b) We write \(\lim _{x \rightarrow a}\) f(x) = -∞ if for a given m < 0, there exists δ > 0 such that |x – a| < δ ⇒ f(x) < m for large |m|.

(c) \(\lim _{x \rightarrow ∞}\) f(x) = l if for given ∈ > 0 there exists k > 0 such that x > k ⇒ |f(x) – l| < ∈ for large x.

(d) \(\lim _{x \rightarrow -∞}\) f(x) = l if for given ∈ > 0, there exists k < 0 such that x < k ⇒ |f(x) – l| < ∈ for large |k|.

(e) We write \(\lim _{x \rightarrow ∞}\) f(x) = ∞ if for m > 0 there exists k > 0 such that x > x ⇒ f(x) > m for large m.

(f) \(\lim _{x \rightarrow ∞}\) x^{n} = \(\left\{\begin{array}{lll}

\infty & \text { if } & n>0 \\

1 & \text { if } & n=0 \\

0 & \text { if } & n<0

\end{array}\right.\)

(g) \(\lim _{n \rightarrow ∞}\) x^{n} = \(\left\{\begin{array}{ccc}

0 & \text { if } & |x|<1 \\

1 & \text { if } & x=1 \\

\infty & \text { for } & x>1

\end{array}\right.\) does not exist for x ≤ -1.

Some useful expansions:

Techniques to find limit:

If \(\lim _{x \rightarrow a}\) f(x), does not take any indeterminate form then get the limit just by putting x = a(provided that the limit is finite).

If \(\lim _{x \rightarrow a}\) f(x) takes any indeterminate form then either use formula or simplify to remove the indeterminate form before finding limit.

The indeterminate form can be removed by using.

- Factorisation
- Rationalisation
- Expand formula or any other techniques.

**Differentiation:**

(a) Let y = f(x) is a function.

The derivative (differential coefficient) of y or f(x) with respect to x is \(\frac{d y}{d x}\) = f'(x) = \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\)

(b) The differentiation of y = f(x) at x = a is \(\left.\frac{\mathrm{dy}}{\mathrm{dx}}\right]_{\mathrm{x}=\mathrm{a}}\) = f'(a) = \(\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}\)

(c) Differentiability of y = f(x) at x = a:

(d) Geometrical meaning of differentiation:

Geometrically f'(x) or \(\frac{d y}{d x}\) represents the slope of tangent to y = f(x) at any point P(x, y)

⇒ Slope of tangent to y = f(x) at A(x_{1}, y_{1}) = \(\left.\frac{d y}{d x}\right]_{\left(x_1, y_1\right)}\)

(e) Some rules of differentiation:

(f) Differentiation of some standard functions: