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Limits Cheat Sheet

 

Limit Properties

If the limit of f(x), and g(x) exists, then the following apply:
limxa(x)=a
limxa[c·f(x)]=c·limxaf(x)
limxa[(f(x))c]=(limxaf(x))c
limxa[f(x)±g(x)]=limxaf(x)±limxag(x)
limxa[f(x)·g(x)]=limxaf(x)·limxag(x)
limxa[f(x)g(x) ]=limxaf(x)limxag(x) ,    where limxag(x)0


Limit to Infinity Properties

For limxcf(x)=,limxcg(x)=L, the following apply:
limxc[f(x)±g(x)]=
limxc[f(x)g(x)]=,    L>0
limxc[f(x)g(x)]=,    L<0
limxcg(x)f(x) =0
limx(axn)=,    a>0
limx(axn)=,    n is even,    a>0
limx(axn)=,    n is odd,    a>0
limx(cxa )=0


Indeterminate Forms

00 0
  00 
0·
1


Common Limits

limx((1+kx )x)=ek limx((xx+k )x)=ek
limx0((1+x)1x )=e


Limit Rules

Limit of a constant limxac=c
Basic Limit limxax=a
Squeeze Theorem
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, limxcf(x)=limxcg(x)=L
Then for any acb, limxch(x)=L
L'Hopital's Rule
For limxa(f(x)g(x) ),
if limxa(f(x)g(x) )=00  or limx a(f(x)g(x) )=±± , then
limxa(f(x)g(x) )=limxa(f(x)g(x) )
Divergence Criterion
If two sequences exist,
{xn}n=1 and {yn}n=1 with
xnc and ync
limnxn=limnyn=c
limnf(xn)limnf(yn)
Then limx cf(x) does not exist
Limit Chain Rule
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