Limits and L'Hospital's Rule

1. L'Hospital's Rule 1 ${\textstyle \lim _{x\to 0}{f(x) \over g(x)}=lim_{x\to 0}{f'(x) \over g'(x)}}$
2. L'Hospital's Rule 2 ${\textstyle \lim _{x\to \infty }{f(x) \over g(x)}=lim_{x\to \infty }{f'(x) \over g'(x)}}$

there are some of the problems that can't use the L'Hospital's Rule such as:

${\textstyle \lim _{x\to \infty }{x+cosx \over x}}$

If you are only using the L'Hospital's Rule this is what you will get:

Start ${\textstyle \lim _{x\to \infty }{x+cosx \over x}}$

1. Derivatives${\textstyle \lim _{x\to \infty }{1-sinx \over 1}}$
2. Derivatives${\textstyle \lim _{x\to \infty }{-cosx}}$
3. Derivatives${\textstyle \lim _{x\to \infty }{sinx}}$

And then so on and forth so that means the limit does not exist.

So if you wan't to compute this first limit we can't use L'Hospital's Rule and this is how we do it.

Start ${\textstyle \lim _{x\to \infty }{x+cosx \over x}}$

1. ${\textstyle \lim _{x\to \infty }{1+{cos \over x}}}$