Evaluate cosx 1 x224x4 Hint Using power series Solutioncos

Evaluate cos(x) - 1 + x2/2/4x4 Hint: Using power series.

Solution

cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ...

Thus, lim x -> 0 (cos (x) - 1 + x²/2)/4x⁴ = lim x -> 0 (1 - x²/2! + x⁴/4! - x⁶/6! + . - 1 + x²/2)/4x⁴ =

lim x -> 0  (x⁴/4! - x⁶/6! ...)/4x⁴ = lim x -> 0 1/(4*4!) - x²/(4*6!) ...= 1/96

You can also solve this using l\'Hopital\'s rule 4 times. You get cos(x)/96, which has the same limit, 1/96.