B - Prírodovedecká fakulta - Univerzita Komenského
B - Prírodovedecká fakulta - Univerzita Komenského
B - Prírodovedecká fakulta - Univerzita Komenského
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Učebné texty z matematiky pre 1. ročník geológie<br />
h.<br />
i.<br />
j.<br />
k.<br />
l.<br />
68<br />
x<br />
lim<br />
x→1<br />
lim<br />
x 1 x<br />
3<br />
2<br />
− 4x<br />
+ x − 2<br />
= lim<br />
x −1<br />
x→1<br />
x −1<br />
= lim<br />
− 3x<br />
+ 2 x→1<br />
2 ( x −1)<br />
( x − 2)<br />
x −1<br />
x −1<br />
( x −1)(<br />
x − 2)<br />
= lim<br />
→ 2<br />
x→1<br />
x<br />
lim<br />
x 2 x<br />
x − 2<br />
= lim<br />
− 5x<br />
+ 6 x→2<br />
x − 2<br />
( x − 2)(<br />
x − 3)<br />
= lim<br />
→ 2<br />
x→2<br />
x<br />
x − 4<br />
lim = lim<br />
x 4 2<br />
x −16<br />
x→4<br />
x − 4<br />
= lim<br />
= lim<br />
x→1<br />
[ ( x −1)(<br />
x − 2)<br />
] = 0<br />
1<br />
= −1<br />
− 2<br />
1<br />
−1<br />
− 3<br />
1<br />
=<br />
( x − 4)(<br />
x + 4)<br />
x→4<br />
+ 4 8<br />
→ x<br />
x + 1<br />
lim = lim<br />
x −1<br />
2<br />
x −1<br />
x→−1<br />
x + 1<br />
= lim<br />
( x + 1)(<br />
x −1)<br />
x→−1<br />
−1<br />
2<br />
→ x<br />
2<br />
x + 20x<br />
+ 100<br />
m. lim<br />
= lim 1 = 1<br />
x→−10 2<br />
x + 20x<br />
+ 100 x→−10<br />
n.<br />
x + 6 x + 6 1 1<br />
lim = lim = lim = −<br />
x→−6<br />
2<br />
x + 6x<br />
x→−6<br />
x<br />
x<br />
100<br />
x<br />
o. lim lim1<br />
= 1<br />
a.<br />
b.<br />
=<br />
x→0 100<br />
x x→0<br />
1<br />
1<br />
( x + 6)<br />
x→−6<br />
6<br />
1<br />
= −<br />
Príklad 39. Vypočítajte nasledovné limity:<br />
tanϕ<br />
lim<br />
ϕ →π<br />
sinϕ<br />
cotgϕ<br />
lim<br />
ϕ →π<br />
/ 2 cosϕ<br />
Riešenie.<br />
tanϕ<br />
sinϕ<br />
1<br />
a. lim = lim = lim = −1<br />
ϕ →π sinϕ<br />
ϕ →π<br />
sinϕ.<br />
cosϕ<br />
ϕ →π<br />
cosϕ<br />
cotgϕ<br />
co sϕ<br />
1<br />
b. lim = lim<br />
= lim = 1<br />
ϕ →π / 2 cosϕ<br />
ϕ →π<br />
/ 2 cosϕ.<br />
sinϕ<br />
ϕ →π<br />
/ 2 sinϕ<br />
2<br />
ln x<br />
c. lim = lim ln x = 0<br />
x→1<br />
ln x x→1<br />
c.<br />
2<br />
ln<br />
lim<br />
x→1 ln<br />
ln<br />
lim<br />
ln<br />
d. + 2<br />
x→0<br />
x<br />
x<br />
x<br />
x