Limite de Trigonométricos

s lim x1 cos πx21x c lim x1 1x tg πx2 d lim x0 cotg 2x cotg π2 x s lim x1 cos πx21x Para x1 temos 00 Mudança de variável tx1 x1 t0 lim t0 cosπ2 π2 tt pois π2 t1 π2 π2 t Sabendo que cosabcos a cos b sen a sen b cos π2 π2 t cos π2 cos π2 t sen π2 sen π2 t 0 1 sen π2 t Para ângulos pequenos de t sen t t quando t tende a 0 Assim sen π2 t π2 t A ssa cosπ2 π2 t π2 t Voltando ao limite lim t0 π2 tt lim t0 π2 π2 c lim 1x tgπx2 x1 1x 0 x 1 πx2 π2 x1 Mudança de variável t1x x1 t0 πx2 π2 π2 t tgπ2 θ cotgθ cosθsenθ θ pequeno senθ θ cosθ 1 θ0 cotgθ cosθsenθ 1θ tgπ2 θ 1θ tgπ2 π2 t 1π2 t 2π t 1x tgπx2 t 2π t 2π lim x1 1x tgπx2 2π d lim x0 cotg2x cotgπ2 x2 cotgπ2 x tgx senxcosx cotg2x cotgπ2 x cos2xsen2x senxcosx Sabemos que cos2x 1 2 sen²x sen2x 2 senx cosx 1 2 sen²x 2 senx cosx senxcosx 1 2 sen²x 2 cos²x x 0 sen0 0 cos0 1 1 2 0 2 1² 12 Substituindo lim x0 cotg2x cotgπ2 x2 12