שליפים עבור גבולות
If the limit of f(x), and g(x) exists, then the following apply:
limx→a(x)=a
limx→a[c·f(x)]=c·limx→af(x)
limx→a[(f(x))c]=(limx→af(x))c
limx→a[f(x)±g(x)]=limx→af(x)±limx→ag(x)
limx→a[f(x)·g(x)]=limx→af(x)·limx→ag(x)
limx→a[f(x)g(x) ]=limx→af(x)limx→ag(x) , where limx→ag(x)≠0
For limx→cf(x)=∞,limx→cg(x)=L, the following apply:
limx→c[f(x)±g(x)]=∞
limx→c[f(x)g(x)]=∞, L>0
limx→c[f(x)g(x)]=−∞, L<0
limx→cg(x)f(x) =0
limx→∞(axn)=∞, a>0
limx→−∞(axn)=∞, n הוא זוגי, a>0
limx→−∞(axn)=−∞, n הוא אי זוגי, a>0
limx→∞(cxa )=0
limx→∞((1+kx )x)=ek
limx→∞((xx+k )x)=e−k
limx→0((1+x)1x )=e
גבול של קבוע
limx→ac=c
גבולות נפוצים
limx→ax=a
משפט הסנדוויץ
Let f, g and h be functions such that for all x∈[a,b] (except possibly at the limit point c),
f(x)≤h(x)≤g(x)
Also suppose that, limx→cf(x)=limx→cg(x)=L
Then for any a≤c≤b, limx→ch(x)=L
חוק לופיטל
For limx→a(f(x)g(x) ),
if limx→a(f(x)g(x) )=00 or limx→ a(f(x)g(x) )=±∞±∞ , then
limx→a(f(x)g(x) )=limx→a(f′(x)g′(x) )
קריטריון להתבדרות
If two sequences exist,
{xn}n=1∞ and {yn}n=1∞ with
xn≠c and yn≠c
limn→∞xn=limn→∞yn=c
limn→∞f(xn)≠limn→∞f(yn)
Then limx→ cf(x) does not exist
חוק השרשרת
if limu → b f(u)=L, and limx → ag(x)=b, and f(x) is continuous at x=b
Then: limx → a f(g(x))=L
אלגברה
גבולות
