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