Section 6.8: Indeterminate Limits

L’Hopital’s Rule

Questions to answer and submit

If the limit is determinate, give its value. If it is indeterminate use L’Hopital’s rule to find the limit.

(lim_(x->2))((sin(2-x))/((x^2)-3x+2))

(lim_(t->-1))(((t^2)-1)/(cos(π t)))

(lim_(y->∞))((2^x)/((x^2)+4x))

Other indeterminate forms

The limits of the form (∞/∞) and (0/0) are both indeterminate. L’Hopital’s rule can be used on both.

Questions to answer and submit

A limit of the form small×large or 0·∞ is also indeterminate. What is the standard trick to change an indeterminate form of this type to one in the form (0/0) or (∞/∞)? Use it to evaluate

(lim_(x->∞))x sin((2/x))

(lim_(t->(SuperPlus[0])))x ln(x)

Questions to answer and submit

Limits of the form (0^0), (1^∞), (∞^0) are also indeterminate forms. Evaluate these by changing to base e.

Write (x^x) with base e. In other words (x^x)=(e^(?)). What is its domain?

Evaluate (lim_(x->a))(x^x) and (lim_(x->b))(x^x) where a and b are the endpoints of the domain.

Questions to answer and submit

The limit
	(lim_(x->∞))(((1+((r )/x)))^x)
is important in many financial applications for fixed values of r.

Is it determinate or indeterminate?

Does the limit exist? If so, what is it?

Questions to answer and submit

Consider the function g(x) = (((x + (e^x)))^(1/x)). It is defined on (0,∞).

What is (lim_(x->(SuperPlus[0])))g(x)?

What is (lim_(x->∞))g(x)?