F i n d space t h e space v a l u e space o f space limit as x rightwards arrow infinity of open parentheses sin 1 over x plus cos 1 over x close parentheses to the power of x

Asked by Topperlearning User | 10th Nov, 2016, 03:16: AM

Expert Answer:

limit as x rightwards arrow infinity of open parentheses sin space 1 over x plus cos space 1 over x close parentheses to the power of x
L e t space y equals 1 over x. space S o space x rightwards arrow infinity comma space t h e n space y rightwards arrow 0
equals limit as y rightwards arrow 0 of open parentheses sin space y plus cos space y close parentheses to the power of 1 over y end exponent
equals e to the power of limit as y rightwards arrow 0 of fraction numerator log space open parentheses sin space y plus cos space y close parentheses over denominator y end fraction space space space open square brackets 0 over 0 space f o r m close square brackets end exponent
equals e to the power of limit as y rightwards arrow 0 of fraction numerator begin display style fraction numerator left parenthesis cos space y minus sin space y right parenthesis over denominator open parentheses sin space y plus cos space y close parentheses end fraction end style over denominator 1 end fraction open square brackets L apostrophe H o s p i t a l apostrophe s space R u l e space close square brackets space space end exponent
equals e to the power of limit as y rightwards arrow 0 of fraction numerator left parenthesis cos space y minus sin space y right parenthesis over denominator open parentheses sin space y plus cos space y close parentheses end fraction end exponent
equals e to the power of limit as y rightwards arrow 0 of fraction numerator left parenthesis 1 minus 0 right parenthesis over denominator open parentheses 0 plus 1 close parentheses end fraction end exponent equals e to the power of 1 equals e

Answered by  | 10th Nov, 2016, 05:16: AM