L.27H.C3.B4pital.27s rule Limit of a function

this rule uses derivatives find limits of indeterminate forms 0/0 or ±∞/∞, , applies such cases. other indeterminate forms may manipulated form. given 2 functions f(x) , g(x), defined on open interval containing desired limit point c, if:

then:

lim

x
→
c





f
(
x
)


g
(
x
)



=

lim

x
→
c






f
′

(
x
)



g
′

(
x
)





{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f (x)}{g (x)}}}

normally, first condition important one.

for example:




lim

x
→
0





sin
⁡
(
2
x
)


sin
⁡
(
3
x
)



=

lim

x
→
0





2
cos
⁡
(
2
x
)


3
cos
⁡
(
3
x
)



=



2
⋅
1


3
⋅
1



=


2
3


.


{\displaystyle \lim _{x\to 0}{\frac {\sin(2x)}{\sin(3x)}}=\lim _{x\to 0}{\frac {2\cos(2x)}{3\cos(3x)}}={\frac {2\cdot 1}{3\cdot 1}}={\frac {2}{3}}.}