1. Teorema Limit Fungsi Trigonometri a. Limit Fungsi Cosinus i. lim !→! cos = 1 Bukti : lim !→! cos = lim !→! cos 0 = lim !→! 1 = 1 ii. lim !→! ! !"# ! = 0 Bukti : lim !→! ! !"# ! = lim !→! ! !"# ! = lim !→! ! !"# ! = ! ! = 0 iii. lim !→! !"# ! ! = ∞ Bukti : lim !→! !"# ! ! = lim !→! !"# ! ! = ! ! = ∞ b. Limit Fungsi Sinus i. lim !→! sin = 0 Bukti : lim !→! sin = lim !→! sin 0 = 0
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1. Teorema Limit Fungsi Trigonometri a. Limit Fungsi Cosinus
i. lim!→! cos 𝑥 = 1 Bukti : lim!→! cos 𝑥 = lim!→! cos 0
= lim!→! 1= 1
ii. lim!→!!
!"#!= 0
Bukti : lim!→!
!!"#!
= lim!→!!
!"#!
= lim!→!!
!"# !
= !!
= 0
iii. lim!→!
!"#!!
= ∞ Bukti : lim!→!
!"#!!
= lim!→!!"# !!
= !!
= ∞
b. Limit Fungsi Sinus i. lim!→! sin 𝑥 = 0 Bukti : lim!→! sin 𝑥 = lim!→! sin 0
= 0
ii. lim
!→!
!!"#!
= 1
Bukti : lim!→!
!!"#!
= lim!→!
!!"# !
= !!
Karena dengan substitusi langsung hasilnya bentuk tak tentu !
! maka
digunakan cara lain di bawah ini
Gambar 3 Pada gambar 3, jika 𝑥 → 0 maka
sin 𝑥 < 𝑥 < tan 𝑥!"#!!"#!
< !!"#!
< !"#!!"#!
1 < !!"#!
< !!"#!
lim!→! 1 < lim!→!
!!"#!
< lim!→!
!!"#!
1 < lim!→!
!!"#!
< lim!→!
!!"# !
1 < lim!→!
!!"#!
< lim!→!
!!
1 < lim!→!
!!"#!
< 1
Sehingga lim
!→!
!!"#!
= 1
iii. lim!→!
!"#!!= 1
Bukti : lim!→!
!"#!!
= lim!→!
!!
!"#!
=!"#!→!
!
!"# !!"#!
!→!
= !!
= 1
iv. lim!→!
!!"#!"
= !!
Bukti : lim!→!
!!"#!"
= lim!→!
!!"#!"
× !!
= lim!→!
!"!"#!"
× !!
= 1× !!
= !!
v. lim!→!
!"!"#!
= 𝑎 Bukti : lim!→!
!"!"#!
= lim!→!
!!"#!
× !!
= lim!→!
!!"#!
×𝑎
= 1×𝑎= 𝑎
vi. lim
!→!
!"!"#!"
= !!
Bukti : lim!→!
!"!"#!"
= lim!→!
!"!"#!"
× !!
= lim!→!
!"!"#!"
× !!
= 1× !!
= !!
c. Limit Fungsi Tangen
i. lim!→! tan 𝑥 = 0 Bukti : lim!→! tan 𝑥 = lim!→! tan 0
= 0
ii. lim!→!!"#!!
= 1
Bukti :
lim!→!!"#!!
= lim!→!
!"#!!"#!!
= lim!→!!"#!! !"#!
= lim!→!!"#!!×lim!→!
!!"#!
= 1×1lim!→!
!"#!!
= 1
iii. lim!→!!
!"#!= 1
Bukti : lim!→!
!!"#!
= lim!→!
!!"#!!
=!"#!→!
!
!"# !"#!!!→!
= !!
= 1
iv. lim!→!
!"#!"!
= 𝑎 Bukti : lim!→!
!"#!"!
= lim!→!!"#!"!
× !!
= lim!→!!"#!"!"
× !!
= 1× !!
= 𝑎
v. lim!→!!"#!!"
= !!
Bukti : lim!→!
!"#!!"
= lim!→!!"#!!"
× !!
= lim!→!!"#!!× !!
= 1× !!
= !!
vi. lim!→!!"#!"!"
= !!
Bukti : lim!→!
!"#!"!"
= lim!→!!"#!"!"
× !!
= lim!→!!"#!"!"
× !!
= 1× !!
= !!
Related Documents
