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)

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