In the limit #lim 10x=40# as #x->4#, how do you find #delta>0# such that whenever #0<abs(x-4)<delta#, #abs(10x-40)<0.01#?

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Answer 1 · 2017-02-28T16:50:56

#0 lt delta le 0.001#.

#|10x-40| lt 0.01,#

#iff |10(x-4)| lt 0.01,#

#iff |10||x-4| lt 0.01,...[because, |ab|=|a||b|]#

#iff 10|x-4| lt 0.01,#

#iff |x-4| lt 0.01/10=0.001,#

Comparing with, #|x-4| lt delta,# we find that,

#0 lt delta le 0.001#.

Observe that, in the event that, #0 lt delta le 0.001,#

#0 lt |x-4| lt delta le 0.001 rArr |10x-40| lt 0.01#.

Thus, the reqd. #delta" is given by, "0 lt delta le 0.001#.

Enjoy Maths.!