462
Chapter 4 | Applications of Derivatives
⎛ ⎝ 1
⎞ ⎠ .
Evaluate lim x →0 +
− 1 tan x
x 2
Solution By combining the fractions, we can write the function as a quotient. Since the least common denominator is x 2 tan x , we have 1 x 2 − 1 tan x = (tan x )− x 2 x 2 tan x . As x →0 + , the numerator tan x − x 2 →0 and the denominator x 2 tan x →0. Therefore, we can apply L’Hôpital’s rule. Taking the derivatives of the numerator and the denominator, we have
⎛ ⎝ sec 2 x ⎞ ⎠ −2 x x 2 sec 2 x +2 x tan x .
(tan x )− x 2 x 2 tan x
lim x →0 +
= lim
x →0 +
⎛ ⎝ sec 2 x ⎞ ⎠ −2 x →1 and x 2 sec 2 x +2 x tan x →0. Since the denominator is positive as x
As x →0 + ,
approaches zero from the right, we conclude that
⎛ ⎝ sec 2 x ⎞ ⎠ −2 x x 2 sec 2 x +2 x tan x
lim x →0 +
=∞.
Therefore,
⎛ ⎝ 1
⎞ ⎠ =∞.
lim x →0 +
− 1 tan x
x 2
Evaluate lim x →0 + ⎛ ⎝ 1
⎞ ⎠ .
4.41
x − 1
sin x
Another type of indeterminate form that arises when evaluating limits involves exponents. The expressions 0 0 , ∞ 0 , and 1 ∞ are all indeterminate forms. On their own, these expressions are meaningless because we cannot actually evaluate these expressions as we would evaluate an expression involving real numbers. Rather, these expressions represent forms that arise when finding limits. Now we examine how L’Hôpital’s rule can be used to evaluate limits involving these indeterminate forms. Since L’Hôpital’s rule applies to quotients, we use the natural logarithm function and its properties to reduce a problem evaluating a limit involving exponents to a related problem involving a limit of a quotient. For example, suppose we want to evaluate lim x → a f ( x ) g ( x ) and we arrive at the indeterminate form ∞ 0 . (The indeterminate forms 0 0 and 1 ∞ can be handled similarly.) We proceed as follows. Let y = f ( x ) g ( x ) . Then, ln y = ln ⎛ ⎝ f ( x ) g ( x ) ⎞ ⎠ = g ( x )ln ⎛ ⎝ f ( x ) ⎞ ⎠ . Therefore, lim x → a ⎡ ⎣ ln( y ) ⎤ ⎦ = lim x → a ⎡ ⎣ g ( x )ln ⎛ ⎝ f ( x ) ⎞ ⎠ ⎤ ⎦ .
f ( x ) =∞, we know that lim x → a ln ⎛
⎞ ⎠ =∞. Therefore, lim x → a
g ( x )ln ⎛
⎞ ⎠ is of the indeterminate form
⎝ f ( x )
⎝ f ( x )
Since lim x → a
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