Limits

A number ‘l’ is called limit of a function f(x) when $x \to a$ i.e., $\overset{lim}{x \to a} f (x) = l$ if given $\epsilon > 0$ , there exists $\delta > 0$ such that |x –a| $< \delta \to$ | f(x) – l | <$\epsilon$ .

Right hand and left hand limits

Let h be a small positive number. Left hand side limit of f(x) when $x \to a$ , is denoted by f(a -0) and is defined as:

$F ( a - 0 ) = \underset{h \to 0}{lim} f (a - h)$

Right hand side limit of f(x), when $x \to a$ , is denoted by f(a + 0) and is defined as:

$f ( a + 0) = \underset{h \to 0}{lim} f(a + h)$

$\underset{x \to a}{lim} f (x)$ exists if

$\underset{h \to 0}{lim} f ( a + h ) = \underset{h \to 0}{lim} f ( a - H)$

Indeterminate forms

If a function f(x) takes the form $f (x) = \dfrac{0}{0} \, \, at \, \, x = a$ , then say that f(x) is indeterminate at x=a. Other Indeterminate Forms are $\dfrac{ \infty}{ \infty} , \infty - \infty , 0^0 , \infty ^0 , 1^{ \infty} , 0 \times \infty$ .

L’ hospital’s rule

If $\phi (x)$ and $\psi (x)$ are functions of x such that $\phi (a) = 0 = \psi (a)$ , then

$\underset{x \to a}{lim} \dfrac{ \phi (x)}{ \phi (x)} = \underset{x \to a}{lim} \dfrac{\phi' (x)}{\psi' (x)}$

The form $0 \times \infty$

This form can easily be reduced either to form $\dfrac{0}{0}$ of $\dfrac{ \infty}{ \infty}$ .

Example:

Evaluate $\underset{x \to 1}{lim} \, \, sec ( \dfrac{ \pi r}{2} ) log x$

Solution:

$= \underset{x \to 1}{lim} \, \, \, sec ( \dfrac{ \pi x}{2} ) log x$

$= \underset{x \to 1}{lim} ( \dfrac{ log x}{ cos ( \pi x / 2} )$

$= \underset{x \to 1}{lim} \dfrac{1 / x}{- \dfrac{ \pi}{2} sin ( \dfrac{ \pi x}{2} )} = - \dfrac{2}{\pi}$

The form $\infty - \infty$

This can also be reduced to the form $\dfrac{0}{0} \, or \, \dfrac{ \infty}{\infty}$

Example:

Evaluate $\underset{x \to 0}{lim} ( \dfrac{1}{x^2} - \dfrac{1}{sin ^2 x} )$

$= \underset{x \to 0}{lim} \dfrac{sin ^2 x - x^2}{x ^2 sin ^2 x}$

$= \underset{x \to 0}{lim} \dfrac{sin ^2 x - x^2}{x^4} ( \dfrac{1}{1} ) ^2$

$= \underset{x \to 0}{lim} \dfrac{ sin (2x) - 2x}{4x^3}$

$= \underset{x \to 0}{lim} \dfrac{ [ 2x - \dfrac{2x^3}{3 !} + \dfrac{ 2x ^5}{5 !} - \cdots ] - 2x}{4x^3}$

$= - \dfrac{8}{3 !} \dfrac{1}{4} = - \dfrac{1}{3}$

Sandwich Theorem (or Squeeze principle)

If f, g, h are functions such that $f (x) \leq g (x) \leq h (x)$ for all x in the neighborhood of a and if $\underset{x \to a}{lim} f (x) = l$ , $\underset{x \to a}{lim} h (x) = l , \underset{x \to a}{lim} g (x) = l$

Algebra of limits

$if \, \, \, \underset{x \to a}{lim} f (x) = l$ ,

$\underset{x \to a} g (x) = m , \, \, then$

$( 1 ) \, \, \, \underset{x \to a}{lim} [ f (x) \pm g (x) ] = \underset{x \to a}{lim} f (x) \pm \underset{ x \to a}{lim} g (x) = l \pm m$

$(2) \, \, \, \underset{x \to a}{lim} f (x) g(x) = \underset{x \to a}{lim} f(x) \underset{x \to a }{lim} g (x) = lm$

$(3) \underset{x \to a}{lim} \dfrac{f (x)}{g (x)} = \dfrac{ \underset{x \to a}{ lim} f (x)}{ \underset{x \to a}{lim} g(x)} = \dfrac{l}{m}$

$(4) [ \underset{x \to a}{lim} f(x) ] ^n = l^n \, \, if \, \, n > 0$

Evaluation of exponential limits of the form $1^{\infty}$

Result:

(i) If $\underset{x \to a}{lim} f (x) = \underset{x \to a}{lim} g (x) = 0 , \, \, then$

$= \underset{x \to a}{lim} [ 1 + f(x) ]^{1 /g(x)} = e \underset{x \to a}{lim} \dfrac{f(x)}{g(x)}$

(ii) If $\underset{x \to a}{lim} f(x) = 1 , \underset{x \to a}{lim} g (x) = \infty , then$

$= \underset{x \to a}{lim} [ f(x)^{g (x)} = \underset{x \to a}{lim} [ 1 + [ f(x) - 1]^{g(x)}$

$= e \, \, \underset{x \to a}{lim} [ f(x) -1 ] g(x)$

Related posts:

1. Partial Differentiation Defination of Partial Differentiation   If f is a function...
2. Continuity and Differentiability Definition of continuity at a point   A function f(x)...
3. Limit Formulas Limit and continuity Formulas Concept of limit and continuity was...
4. Linear Differential Equations Linear Differential Equation of nth Order Linear differential equation is...
5. Properties of Definite Integral Properties of Definite Integral Definite integral is part of integral...