Topic today: L’Hôpital’s rule
Indeterminant forms: 
.
The idea about dealing with the indeterminant forms 
, is
to rewrite the function as a quotient (that is, into the form of 
or 
).
(1) The 0/0 case:
Typical example: the derivative of a function 
#card=math&code=f%28x%29&id=TsGYN) at 
, that is 
-f(a)%7D%7Bx-a%7D%20%3D%20%5Clim%7Bh%5Cto%200%7D%20%5Cfrac%7Bf(a%2Bh)-f(a)%7D%7Bh%7D#card=math&code=%5Clim%7Bx%20%5Cto%20a%7D%20%5Cfrac%7Bf%28x%29-f%28a%29%7D%7Bx-a%7D%20%3D%20%5Clim_%7Bh%5Cto%200%7D%20%5Cfrac%7Bf%28a%2Bh%29-f%28a%29%7D%7Bh%7D&id=vbHMc).
(2) The 
case: 
.
 (without using L'Hôpital's rule)
Note: Pay attention to the hypotheses on L’Hôpital’s rule (Theorem 4.12, 4.13)!
See the counter-example 4.40 where L’Hôpital’s rule cannot apply
Another counter-example: 
%7D%7Bx%7D%3D0#card=math&code=%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B%5Csin%28x%29%7D%7Bx%7D%3D0&id=OHb1o) . To see this, using the squeeze theorem: 
as 
.
However, (taking derivatives for both numerator and denominator) 
%7D%7B1%7D#card=math&code=%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B%5Ccos%28x%29%7D%7B1%7D&id=cc3PW) DNE. (does not exist).
(3) 
Useful result: Example 4.41 
.
%20%3D%20%5Cfrac%7B%5Ccos(x)%7D%7B%5Csin(x)%7D%2C#card=math&code=%5Ccot%28x%29%20%3D%20%5Cfrac%7B%5Ccos%28x%29%7D%7B%5Csin%28x%29%7D%2C&id=wRbN2) ‘%20%3D%20%5Cfrac%7B-%5Csin%5E2(x)-%5Ccos%5E2(x)%7D%7B%5Csin%5E2(x)%7D%3D-%5Cfrac%7B1%7D%7B%5Csin%5E2(x)%7D%2C#card=math&code=%5Ccot%28x%29%27%20%3D%20%5Cfrac%7B-%5Csin%5E2%28x%29-%5Ccos%5E2%28x%29%7D%7B%5Csin%5E2%28x%29%7D%3D-%5Cfrac%7B1%7D%7B%5Csin%5E2%28x%29%7D%2C&id=OwMYN) 
‘%20%3D%20-%5Csin(x).#card=math&code=%5Ccos%28x%29%27%20%3D%20-%5Csin%28x%29.&id=MJryD)
%7D%20%3D%20%5Clim%7Bx%5Cto%200%2B%7D%5Cfrac%7B1%2Fx%7D%7B-1%2F%5Csin%5E2(x)%7D%20%3D%20%5Clim%7Bx%5Cto%200%2B%7D%20-%5Cfrac%7B%5Csin%5E2(x)%7D%7Bx%7D#card=math&code=%5Clim%7Bx%20%5Cto%200%2B%7D%20%5Cfrac%7B%5Cln%20x%7D%7B%5Ccot%28x%29%7D%20%3D%20%5Clim%7Bx%5Cto%200%2B%7D%5Cfrac%7B1%2Fx%7D%7B-1%2F%5Csin%5E2%28x%29%7D%20%3D%20%5Clim_%7Bx%5Cto%200%2B%7D%20-%5Cfrac%7B%5Csin%5E2%28x%29%7D%7Bx%7D&id=ONR8L)
)%5Ccdot%20%5Cfrac%7B%5Csin(x)%7D%7Bx%7D%20%3D%200%20%5Ccdot%201%3D0#card=math&code=%5Clim_%7Bx%5Cto%200%2B%7D%20%28-%5Csin%28x%29%29%5Ccdot%20%5Cfrac%7B%5Csin%28x%29%7D%7Bx%7D%20%3D%200%20%5Ccdot%201%3D0&id=mEG57)
Remark: another solution:
%2F%5Ccot(x)%20%3D%20%5Clim%7Bx%5Cto%200%2B%20%7D%20%5Ctan(x)%5Ccdot%20%5Cln(x)%20%3D%20%5Clim%7Bx%5Cto%200%2B%7D%20x%20%5Ccdot%20%5Cln(x)%20%3D0#card=math&code=%5Clim%7Bx%5Cto%200%2B%7D%20%5Cln%28x%29%2F%5Ccot%28x%29%20%3D%20%5Clim%7Bx%5Cto%200%2B%20%7D%20%5Ctan%28x%29%5Ccdot%20%5Cln%28x%29%20%3D%20%5Clim_%7Bx%5Cto%200%2B%7D%20x%20%5Ccdot%20%5Cln%28x%29%20%3D0&id=PMbnI) (by Example 4.41)
Note: very useful (and hence important) limits:
(just write 
%20%3D%20%5Csin(%5Ctheta)%2F%5Ccos(%5Ctheta)#card=math&code=%5Ctan%28%5Ctheta%29%20%3D%20%5Csin%28%5Ctheta%29%2F%5Ccos%28%5Ctheta%29&id=ckCak))
(4) 
Examle 4.42.
-x%5E2%7D%7Bx%5E2%20%5Ctan(x)%7D%20%3D%20%5Clim%7Bx%5Cto%200%2B%7D%20%5Cfrac%7B%5Ctan(x)-x%5E2%7D%7Bx%5E3%7D%5Cfrac%7Bx%7D%7B%5Ctan(x)%7D#card=math&code=%5Clim%7Bx%20%5Cto%200%2B%7D%20%5Cfrac%7B%5Ctan%28x%29-x%5E2%7D%7Bx%5E2%20%5Ctan%28x%29%7D%20%3D%20%5Clim%7Bx%5Cto%200%2B%7D%20%5Cfrac%7B%5Ctan%28x%29-x%5E2%7D%7Bx%5E3%7D%5Cfrac%7Bx%7D%7B%5Ctan%28x%29%7D&id=oOyHj) and 
Basic formula for logarithm: 
(5) Exponential forms 
Example 4.43: 
Its discrete version: 
Example 4.44: 
What is 
#card=math&code=%5Ccsc%28x%29&id=DWjue)? What is its derivative?
%3D1%2F%5Csin(x)#card=math&code=%5Ccsc%28x%29%3D1%2F%5Csin%28x%29&id=Rl77T) . 
%20%5Ccdot%20%5Ccot(x)%20%3D%20-%5Cfrac%7B1%7D%7B%5Csin(x)%20%7D%20%5Cfrac%7B%5Ccos(x)%7D%7B%5Csin(x)%7D#card=math&code=-%5Ccsc%28x%29%20%5Ccdot%20%5Ccot%28x%29%20%3D%20-%5Cfrac%7B1%7D%7B%5Csin%28x%29%20%7D%20%5Cfrac%7B%5Ccos%28x%29%7D%7B%5Csin%28x%29%7D&id=lQGHq)
%20%5Cln(x)%20%3D%20%5Clim%7Bx%20%5Cto%200%2B%7D%20%5Cfrac%7B%5Csin(x)%7D%7Bx%7D%20%5Ccdot%20x%20%5Cln(x)%20%3D%201%5Ccdot%200%3D0#card=math&code=%5Clim%7Bx%5Cto%200%2B%7D%20%5Csin%28x%29%20%5Cln%28x%29%20%3D%20%5Clim_%7Bx%20%5Cto%200%2B%7D%20%5Cfrac%7B%5Csin%28x%29%7D%7Bx%7D%20%5Ccdot%20x%20%5Cln%28x%29%20%3D%201%5Ccdot%200%3D0&id=BQNaL).
Remark: It is important to simplify when using L’Hôpital’s rule, for instance, the trick of “Equivalent Infinitesimal Replacement” should be kept in mind.
Optional: watch how to apply derivatives to solve optimization problems through Khan academy https://www.khanacademy.org/math/old-differential-calculus/derivative-applications-dc/optimization-dc/v/optimizing-box-volume-graphically
