Limites

3. limt →π4 [tg9tt-π4, 5t-π4-1t-π4 , ln⁡(etg t)]
limx →0tg9x+π4 -1x=limx →01xtg9(x)+tg9(π4 )1+tg9(x)tg9(π4 )-1 =limx →01xtg9(x)1+tg9(x)-1=limx →01x2tg9(x)1+tg9(x)= limx →0sen9xx2cos9x(1+tg9x)=1.2.1=2
limt →π4 5t-π4-1t-π4=5π4-π4-1π4-π4= 1-10= 00
limt →π4 lnetg t=lnetg π4= lne1=1
limt →π4 [tg9tt-π4, 5t-π4-1t-π4 , ln⁡(etg t)]=( 2, , 1 )
4. limt→0 [tsenπ4t , e4t + e-7t4+ t4 ,e10t - e15tsen10t - sen15t]
limt→0tsenπ4t=π4limt→01senπ4tπ4t =π4 .1=π4
limt→0e4t + e-7t4+ t4= e0 + e04+ t4= 24= 12
limt→0e10t - e15tsen10t - sen15t=
5. limt →5 ⁡[5t-5sent2-5tsecπt10t -5 , 5t -25sen5tπ]
lim⁡ t →55t-5sent2-5tsecπt10t - 5 =0-5sen0-0sec00 - 5=0 - 5 =0
limt →5 5t -25sen5tπ= 25 -25sen0π =
6. limt →0 [1 - cos6t2t2 , lnt+4-ln4t , 5t-3t7t-4t]
limt →0 1 -cos6t2t2= 1 - cos60202= 00
lim⁡t →0lnt+4-ln4t= ln t+ln4-ln4t=lim⁡t →0lntt=1
limt →05t-3t7t-4t= tln5-tln3tln7-tln4=limt →0ln5-ln3ln7-ln4=ln53ln74
7.limt →0 ⁡[5sen t-3cos t +32tg t +1-cost , t2t+64t22 1-cost , t-sent2t sent 3]
limt →05sen t-3cos t +32tg t + 1-cost =5sen t-3(cos t -1)2tg t +(1-cost) =5sen t(1+cost)-3sen2t2tg t (1+cost) +sen2t =limt →05(1+cost)cost2(1+cost).22=52
limt →0t2t+64t22 1-cost = t2t+64t2(1+cost)2 1-cost(1+cost) = t2t+64t2(1+cost)2sen2t=12.2.0=0
limt →0t-sent2t sent 3=
8. limt →0 [1+sent-1-sent t , sent +sen3t-12sen2tt2 sent ,tgt-sent t -tcost2]
lim⁡ [t →01+sent-1-sent t]=lim⁡ t →01+sent-1-sent t1+sent+1-sent1+sent+1-sent=1+sent-1-sent t(1+sent+1-sent)=2sentt(1+sent+1-sent) =lim⁡ t →0sentt lim⁡ t →021+sent+1-sent=1. 1=1lim⁡ [t →0sent +sen3t-12sen2tt2 sent ]= sent +sen3t-sent.costt2 sent =1t2+sen2t- costt2=1-costt2+ sen2t=lim⁡ [t →01-costt21+cost1+cost]+ lim⁡ [t →0sen2t]= 0+lim⁡ [t →0sen2tt2]lim⁡ [t →011+cost]= 12lim⁡t →0tgt-sent t -tcost2=lim⁡t →0tgt-sentt( 1-cost)=tgt-sent ( 1+cost)t( 1-cost)( 1+cost )= tgt-sent ( 1+cost)tsen2t=sen2tcost t sent= lim⁡t →0 sentt cost=1
limt →0 [1+sent-1-sent t , sent...