Question

help please number 22

http://prnt.sc/aywxcr

Answer 1

This is using L'Hopital's Rule.
To test, sub in the 0 and see you get 0/0 an indeterminate form.
Providing f'(x)/g'(x) has a limit defined as x->0 to be L, then this limit is also the limit of the original quotient.

What happens here is you differentiate both numerator and denominator, sub in 0 and see it is still undefined 0/0...so you keep differentiating until you do get a defined limit, which will be the answer

Answer 2

\[\lim_{x \rightarrow 0}\frac{ f(x) }{ g(x) } = \lim_{x \rightarrow 0}\frac{ f'(x) }{ g'(x) } = \lim_{x \rightarrow 0}\frac{ f''(x) }{ g''(x) } = ... = L \]

you stop when you do get a defined value for L but remember it has to start off as a 0/0 or infinity/infinity (indeterminate form) for you to apply L'Hopital's Rule and f and g must be differentiable at the point.