InterviewSolution
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InterviewSolution
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` lim_(x to 1) sqrt(1-cos 2(x-1))/(x-1)`A. exists and it equals `sqrt2`B. exists and it equals `-sqrt2`C. does not exist because ` x -1 to 0`D. does not exist because left hand limit is not equal to right hand limit |
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Answer» Correct Answer - C LHL `underset(x to 1^(-)) lim sqrt(1-cos 2(x-1))/(x-1)` ` = underset(x to 1^(-)) limsqrt(2 sin^(2)(x-1))/(x-1) = sqrt2 underset(x to 1^(-)) lim (|sin (x-1)|)/(x-1)` Put ` x = 1 - h, h gt 0, " for " x to 1^(-) , h to 0` ` = sqrt2 underset ( h to 0) lim (|sin (-h)|)/(-h) ` ` = sqrt2 underset( h to 0) lim (sin h)/(-h) = - sqrt2` Again, RHL `underset( x to 1^(+)) lim sqrt(1-cos 2 (x -1))/(x-1)` ` underset ( x to 1^(+)) lim sqrt2 (|sin ( x - 1)|)/(x - 1) ` Put ` x = 1 + h, h gt 0 ` For ` x to 1^(+), h to 0` `underset( h to 0) lim sqrt2 (| sin h |)/h = underset( h to 0) lim sqrt2 (sin h)/h = sqrt2 ` ` :. ` LHL `ne` RHL. Hence, `underset( x to 1) lim f (x) ` does not exist. |
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