Discuss the differentiability at x=0 of the funtion **PLEASE SHOW ALL WORK!! Sol

Question

Discuss the differentiability at x=0 of the funtion **PLEASE SHOW ALL WORK!!

Explanation / Answer

we know that the range of sin function is [-1,1]. we prove lim(x-->0) sin ( 1/x) doesnot exist by contradiction. i,.e. suppose exists and = L. if at all lim(x-->0) sin ( 1/x) exists then it must be one among [-1,1]. keeping this in view, we split into two cases. (1) L not equal to 1. keeping the definition of limit in view, suppose to each smallestpositive real number , there exists a > 0such that | x - 0| < ==> | f(x) - L| | 1- L| < 1 -L this is a contradiction. so, L = 1 does not hold. (2) suppose L = 1. then without loss of generality, choose = 1. in this case, we can find the largest natural number n andconsequently x = 1/ n is smallest such that | x - 0 | | 0-L| < 1 ==> 1 < 1 again a contradiction. combining the above cases, we confirm that lim(x-->0) sin ( 1/x) does notexist.

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