28-OCT

Comments on homework problems:

1. Checking whether there is an asymptote or not.

Vertical asymptote: for #card=math&code=f%28x%29): %20%3D%20%5Cpm%20%5Cinfty#card=math&code=%5Clim_%7Bx%20%5Cto%20a%20%5Cpm%7D%20f%28x%29%20%3D%20%5Cpm%20%5Cinfty);

Horizontal asymtote: If %20%3D%20M%20%3C%20%5Cinfty#card=math&code=%5Clim_%7Bx%20%5Cto%20%5Cpm%20%5Cinfty%7D%20f%28x%29%20%3D%20M%20%3C%20%5Cinfty), then is a horizontal asymptote for #card=math&code=f%28x%29).

1. A general hyperbola symmetric about #card=math&code=%28a%2C%20b%29): . Set . .

2. Prob. 356. ( case, using L’Hôpital’s rule. )

(formally)

1. , domain is #card=math&code=%28-%5Cinfty%2C%20%5Cinfty%29) and it is a monotonically increased function over its domain.

As , also goes to infinity.

, ….

$(x2)=(x-a)(x+a), \quad (x3) = (x-a)(x2) $

#card=math&code=%5Clim_%7Bx%20%5Cto%20a%7D%5Cfrac%7B1%7D%7Bx%2Ba%7D%20%3D%201%2F%282a%29)

#card=math&code=%5Clim_%7Bx%20%5Cto%20a%7D%20%5Cfrac%7B1%7D%7Bx%5E2%2Bax%2Ba%5E2%7D%20%3D%201%2F%283a%5E2%29)

Prob. 359 by L’Hôpital’s rule. (it is a 0/0 indeterminate form)

#card=math&code=%5Clim%7Bx%20%5Cto%20a%7D%20%5Cfrac%7Bx-a%7D%7Bx%5E2-a%5E2%7D%20%3D%20%5Clim%7Bx%5Cto%20a%7D%20%5Cfrac%7B1%7D%7B2x%7D%20%3D%201%2F%282a%29)

1. A related rate problem
   , #card=math&code=x%3Dx%28t%29) then #card=math&code=y%3Dy%28t%29) .
   Now . Thus .