फलन ` f ( x ) `, निम्नवत परिभाषित है -
` f ( x ) = {{:( x",",, "यदि " 0 le x lt ( 1 ) /(2)),( 0",",, "यदि " x = ( 1 ) /(2)),( 1-x",",, "यदि " (1)/(2) lt x le 1 ):}`
` lim _ ( x to 1//2) f ( x ) ` का मान ज्ञात कीजिये |

Explanation

फलन की वाम पक्ष सीमा
माना ` x = ( 1 ) /(2 ) - h ` यदि ` x = (1) /(2) rArr h = 0 `
` therefore lim _ ( x to (1) /(2)) rArr lim _ ( h to0) `
` therefore lim _ (x to ((1)/(2))^(-)) f ( x ) = lim _ ( h to 0 ) f (( 1 ) /(2) - h ) = lim _ ( hto 0) [ (1) /(2) - h ] `
` = ((1) /(2) -0) = (1)/(2)" " (because h = 0 ) `
फलन की दक्षिण पक्ष सीमा
माना ` x = (1 )/(2) + h ` यदि ` x = (1)/(2) rArr h = 0 `
` therefore lim _ ( x to ((1 )/(2))^( +)) rArr lim _ ( h to 0)`
` therefore lim _ ( x to ((1)/(2)) ^( + ) ) f ( x ) = lim _ (h to 0) f (( 1 ) /(2) + h ) `
` = lim _ ( h to 0 ) [ 1 - ((1)/(2) + h)] = lim _ ( h to 0) [ (1) /(2) - h] `
` = ((1)/(2) -0) " " ( because h = 0 ) `
` = (1) /(2) `
स्पष्ट है कि फलन की वाम पक्ष सीमा = फलन की दक्षिण पक्ष सीमा ` = (1)/(2) `
` therefore lim _ ( x to (1)/(2)) f ( x ) = (1)/(2) `