lim           x^2(1-cos1/x)  
xrightwards arrowinfinity
find limits in the above question

Asked by JAIBHARGAVA200 | 3rd Nov, 2014, 02:27: PM

Expert Answer:

W e space n e e d space t o space e v a l u a t e space limit as x rightwards arrow infinity of x squared open parentheses 1 minus cos open parentheses 1 over x close parentheses close parentheses W e space k n o w space t h a t comma open parentheses 1 minus cos open parentheses 1 over x close parentheses close parentheses equals 2 sin squared open parentheses fraction numerator 1 over denominator 2 x end fraction close parentheses T h u s comma space limit as x rightwards arrow infinity of x squared open parentheses 1 minus cos open parentheses 1 over x close parentheses close parentheses equals space 2 limit as x rightwards arrow infinity of x squared cross times sin squared open parentheses fraction numerator 1 over denominator 2 x end fraction close parentheses equals space 2 limit as x rightwards arrow infinity of fraction numerator sin squared open parentheses fraction numerator 1 over denominator 2 x end fraction close parentheses over denominator open parentheses begin display style fraction numerator 1 over denominator 2 x end fraction end style close parentheses squared end fraction cross times 1 fourth equals limit as x rightwards arrow infinity of fraction numerator sin squared open parentheses fraction numerator 1 over denominator 2 x end fraction close parentheses over denominator open parentheses begin display style fraction numerator 1 over denominator 2 x end fraction end style close parentheses squared end fraction cross times 1 half L e t space u s space s u b s t i t u t e comma space t equals fraction numerator 1 over denominator 2 x end fraction ; space A s space x rightwards arrow infinity comma t rightwards arrow 0 T h u s comma limit as x rightwards arrow infinity of x squared open parentheses 1 minus cos open parentheses 1 over x close parentheses close parentheses equals limit as t rightwards arrow 0 of fraction numerator sin squared t over denominator t squared end fraction cross times 1 half equals 1 half cross times limit as t rightwards arrow 0 of fraction numerator sin squared t over denominator t squared end fraction equals 1 half cross times limit as t rightwards arrow 0 of fraction numerator sin t over denominator t end fraction cross times limit as t rightwards arrow 0 of fraction numerator sin t over denominator t end fraction equals 1 half cross times 1 cross times 1 space space space space space open square brackets S i n c e comma space limit as t rightwards arrow 0 of fraction numerator sin t over denominator t end fraction equals 1 close square brackets equals 1 half

Answered by Vimala Ramamurthy | 4th Nov, 2014, 09:17: AM