How to solve lim 1-cos(x)/1-cos(4x) as x tends to zero, Mathematics

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Use L''hopital''s rule

since lim X-->0 1-cos(x)/1-cos(4x) is in the indeterminate form 0/0 when we apply the limt

so by l''hoptital''s rule differentiate the numerator and denominatior

so we get lim x-->0 sin(x)/4sin(4x) again in the indeterminate form when we apply limit so again use l''hopitals'' rule

lim x-->0 cos(x)/16cos(4x) now if u apply the limit we get it as 1/16

so lim x--->0 1-cos(x)/1-cos(4x) = 1/16

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