Limits- indeterminate forms and L’Hospital’s Rule

I. Indeterminate Form of the Type

We have previously studied limits with the indeterminate form as shown in the following example:

Example:

However, there is a general, systematic method for determining limits with the indeterminate form . Suppose that f and g are differentiable functions at x = a a nd that is an indeterminate form of the type ; that is, and . Since f and g are differentiable functions at x = a, then f and g are continuous at x = a; that is, = 0 and = 0. Furthermore, since f and g are differentiable functions at x = a, then and . Thus, if , then

if and

are continuous at x = a. This illustrates a special case of the technique known as L’Hospital’s Rule.

L’Hospital’s Rule for Form

Suppose that f and g are differentiable functions on an open interval

containing x = a, except possibly at x = a, and that and

. If has a finite limit, or if this limit is or

, then . Moreover, this statement is also true

in the case of a limit as or as

In the following examples, we will use the following three-step process:

Step 1. Check that the limit of is an indeterminate form of type . If it is not, then L’Hospital’s Rule cannot be used.

Step 2. Differentiate f and g separately. [ Note: Do not differentiate using the quotient rule! ]

Step 3. Find the limit of . If this limit is finite, , or , then it is equal to the limit of . If the limit is an indeterminate form of type , then simplify algebraically and apply L’Hospital’s Rule again.

II. Indeterminate Form of the Type

We have previously studied limits with the indeterminate form as shown in the following example:

Example:

However, we could use another version of L’Hospital’s Rule.

L’Hospital’s Rule for Form

Suppose that f and g are differentiable functions on an open interval

containing x = a, except possibly at x = a, and that and

. If has a finite limit, or if this limit is or

, then . Moreover, this statement is also true

in the case of a limit as or as

III. Indeterminate Form of the Type

Indeterminate forms of the type can sometimes be evaluated by rewriting the product as a quotient, and then applying L’Hospital’s Rule for the indeterminate

forms of type or .

Example:

IV. Indeterminate Form of the Type

A limit problem that leads to one of the expressions

, , ,

is called an indeterminate form of type . Such limits are indeterminate because the two terms exert conflicting influences on the expression; one pushes it in the positive direction and the other pushes it in the negative direction. However, limits problems that lead to one the expressions

, , ,

are not indeterminate, since the two terms work together (the first two produce a limit of and the last two produce a limit of ). Indeterminate forms of the type can sometimes be evaluated by combining the terms and manipulating the result to produce an indeterminate form of type or .

Example:

V. Indeterminate Forms of the Types

Limits of the form frequently give rise to indeterminate forms of the types . These indeterminate forms can sometimes be evaluated as follows:

(1)

(2)

(3)

The limit on the right hand side of the equation will usually be an indeterminate limit of the type . Evaluate this limit using the technique previously described. Assume that = L.

(4) Finally, .

Example: Find .

This is an indeterminate form of the type . Let . 0.

Thus, .

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