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Evaluate: lim(x->0)8/(x^8){1-cos((x^2)/2...

Evaluate: `lim_(x->0)8/(x^8){1-cos((x^2)/2)-cos((x^2)/4)+cos((x^2)/2)cos((x^2)/4)}`

Text Solution

Verified by Experts

The correct Answer is:
`1//32`
`underset(xto0)lim(8(1-"cos"(x^(2))/(4))(1-"cos"(x^(2))/(2)))/(x^(8))`
`=underset(xto0)lim(8xx2"sin"^(2)(x^(2))/(8)xx2"sin"^(2)(x^(2))/(4))/(x^(8))`
`=underset(xto0)lim(32)/(64xx16).("sin"^(2)(x^(2))/(8))/(x^(4)/(64)).(2"sin"^(2)(x^(2))/(4))/(x^(4)/(16))`
`=1/32`
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