Please explain how you got your answer The many variations of L\'Hospital\'s rul
ID: 2970514 • Letter: P
Question
Please explain how you got your answer
The many variations of L'Hospital's rule relate the limit of the quotient of two functions to the corresponding limit of the quotient of the derivatives of the two functions. Some restrictions apply. The following is a sample statement. Suppose that f and g are differentiable functions on a deleted neighborhood I-{c} of c and that limx rightarrow cf(x) = 0 = limx rightarrow cg(x). Suppose also that g'(x) 0 on I - {c} and that limxrightarrwc f'(x) / g' exists Then limx rightarrow c f(x) / g(x) exists and equals xrightarrowcf'(x) / g'(x). Prove this result. (Hint: Consider the one-sided limits limx rightarrow c plusminus f(x) / g(x) separately, using in each case the EMVT, but be cautious about assumptions and division by zero.)Explanation / Answer
let d be a very small positive number
now left hand limit:
as f and g are not defined at c,
we can take left hand limit of lim f/g as x tends to c
as equal to
lim (f(x)/g(x)) as x tends to c-d
=f(c-d)/g(c-d)
=(f(c)-f(c-d))/(g(c)-g(c-d))
f'(x)/g'(x) at x=c
as differentiation is defined as
f'(x) at x=c
is given by
lim (f(c)- f(c-h))/h
with h being very small
so similarly we can prove for right hand limit
and find that limit is given by f'(x)/g'(x) as x tends to c.