L'Hopitals Rule Question

Question

Find the limit using L'Hopital's rule when needed:

lim (csc x - cot x)

x--->0

Answer 1

lim (csc x - cot x) = lim (1 / sin x - cos x / sin x) = lim (1 - cos x ) / sin x = |0 / 0| {use L'Hopital's rule}

x --> 0 x --> 0 x --> 0

Applying L'Hopital's rule, we get:

lim (1 - cos x ) / sin x = {LHR} = lim sin x / cos x = 0

x -->0 x --> 0

Answer 2

1) Through plugging in for the limite, we find out that we have infinity minus infinity form. (Indeterminate)

2) Reexpress the subtraction using trigonometric identities

3) sinx is the common denominator

4) Use L'Hopital's Rule with the derivative of the top over the derivate of the bottom

5) Plug in limit

6) Final Answer